Amplificadores Raman

Universidade de Aveiro Departamento de Física,2010

Berta MariaBarbosa Neto

Técnicas alternativas para amplificação de Ramanem telecomunicações

“Sometimes, the reflection is

far more present than the

thing being reflected.”

— Jim Jarsmusch

in The Limits of Control







Dissertação apresentada à Universidade de Aveiro para cumprimento dosrequesitos necessários à obtenção do grau de Doutor em Física, realizadasob a orientação científica de Paulo Sérgio de Brito André, Investigadorauxiliar do Instituto de Telecomunicações (IT) e Professor auxiliar doDepartamento de Física da Universidade de Aveiro . Apoio financeiro daFCT e do FSE no âmbito do VI Quadro Comunitário de Apoio

o júri / the jury

presidente / president Doutor João Manuel Nunes TorrãoProfessor Catedrático da Universidade de Aveiro (por delegação da Reitora

da Universidade de Aveiro)

vogais / examinerscommittee

Doutor João de Lemos PintoProfessor Catedrático da Universidade de Aveiro

Doutor José Luís Campos de Oliveira SantosProfessor Professor Associado com Agregação da Faculdade de Ciências

da Universidade do Porto

Doutor Hypolito José KalinowskyProfessor Associado da Universidade Tecnológica Federal do Paraná,

Brasil

Doutor Henrique José Almeida SilvaProfessor Associado da Universidade de Coimbra

Doutor António Luís Jesus TeixeiraProfessor Associado da Universidade de Aveiro

Doutor Paulo Sérgio de Brito AndréInvestigador auxiliar do Instituto de Telecomunicações (IT) e Professor

Auxiliar da Universidade de Aveiro e (orientador)

agradecimentos O trabalho desenvolvido no âmbito desta tese contou, felizmente,com as contribuições de pessoas e instituições , pelo que queroexpressar a todos os meus mais sinceros agradecimentos.Ao Doutor Paulo André, pela atitude diligente com que orientouesta tese, nomeadamente o apoio prestado em todas as fases dotrabalho, a disponibilidade para esclarecimento de dúvidas, algumaliberdade de ação que muito contribuiu para o meu desenvolvimentopessoal, e finalmente o apoio na revisão dos vários capítulos destatese.À Fundação para a Ciência e Tecnologia (FCT) pelosuporte específico dado a este trabalho através da bolsa(SFRH/BD/28904/2006) e pelo financiamento providenciado atravésdos projectos ARPA (POSC/EEA-CPS/55781/2004), TECLAR(POSI/V.5/A0072/2005) e FEFOF (PTDC/EEA-TEL/72025/2006).Ao Instituto de Telecomunicações (IT) por ter disponibilizado osrecursos materiais que permitiram a realização desta dissertação.Ao National Intitute of Information and Communications Technology(NICT), Tóquio, Japão, pelo acolhimento e disponibilização derecusos labororatoriais de topo no âmbito de um estágio de curtaduração, ao Doutor Hideaki Furakawa pelo acompanhamentoexperimental e ao Doutor Nayoa Wada pela sua atenta supervisão.Aos Doutores António Teixeira, Rogério Nogueira e NatasaPavlovich, investigadores do IT e da Nokia Siemens Networks, pelapela interação, aprendizagem, por muitas e úteis informações queme ajudaram a perspectivar outros pontos de vista sobre o meutrabalho e pelos constantes incentivos e amizade.Ao Mestre Claunir Pavan pela imprescindível ajuda e apoio coma edição da tese. Aos Mestres Ana Rocha, Donato Sperti,Cláudia Reis, Rogério Dionísio, Jacklyn Reis, Rui Morais, JoãoFerreira, Albano Baptista e Rodolfo Andrade, Engs. Pedro Teixeira,João Andrade, Andreas Klingler e José Pedro Girão, agradeçoa colaboração no trabalho experimental e outras aprendizagensparalelas aos trabalhos desta tese.Finalmente, aos meus colegas do grupo de comunicações óticas,pelo companheirismo e boa disposição com que engrandeceram omeu dia-a-dia.

Palavras Chave Redes de acesso, amplificadores de Raman em fibra ótica,otimização do ganho, redes óticas de rajadas/pacotes, efeitostransitórios

Resumo O presente trabalho centra-se no estudo dos amplificadores deRaman em fibra ótica e suas aplicações em sistemas modernosde comunicações óticas. Abordaram-se tópicos específicos comoa simulação espacial do amplificador de Raman, a equalizaçãoe alargamento do ganho, o uso de abordagens híbridas deamplificação através da associação de amplificadores de Ramanem fibra ótica com amplificadores de fibra dopada com Érbio(EDFA) e os efeitos transitórios no ganho dos amplificadores. Asactividades realizadas basearam-se em modelos teóricos, sendo osresultados validados experimentalmente.De entre as contribuições mais importantes desta tese, destaca-se(i) o desenvolvimento de um simulador eficiente para amplificadoresde Raman que suporta arquitecturas de bombeamentocontraprogantes e bidirecionais num contexto com multiplexagemno comprimento de onda (WDM); (ii) a implementação de umalgoritmo de alocação de sinais de bombeamento usando acombinação do algoritmo genético com o método de Nelder-Mead; (iii) a apreciação de soluções de amplificação híbridas porassociação dos amplificadores de Raman com EDFA em cenáriosde redes óticas passivas, nomeadamente WDM/TDM-PON comextensão a região espectral C+L; e (iv) a avaliação e caracterizaçãode fenómenos transitórios em amplificadores para tráfego emrajadas/pacotes óticos e consequente desenvolvimento de soluçõesde mitigação baseadas em técnicas de clamping ótico.

Key words Optical access networks, Raman fiber amplifiers, gain optimization,optical burst/packet switching networks, transient effect in opticalamplifiers.

Abstract The present work is based on Raman Fiber Amplifiers and theirapplications in modern fiber communication systems. Specific topicswere approached, namely the spatial simulation of Raman fiberamplifiers, the gain enlargement and equalization the use of hybridamplification approaches by association of Raman amplifiers withErbium doped fiber amplifiers (EDFA) and the transient effect onoptical amplifiers gain. The work is based on theoretical models,being the obtained results validated experimentally.Among the main contributions, we remark: (i) the development of anefficient simulator for Raman fiber amplifiers that supports backwardand bidirectional pumping architectures in a wavelength divisionmultiplexing (WDM) context; (ii) the implementation of an algorithmto obtain enlargement and equalization of gain by allocation ofpumps based on the association of the genetic algorithm with theNelder-Mead method; (iii) the assessment of hybrid amplificationsolutions using Raman amplifiers and EDFA in the context of passiveoptical networks, namely WDM/TDM-PON with extension the C+Lspectral bands; (iv) the assessment and characterization of transienteffects on optical amplifiers with bursty/packeted traffic and thedevelopment of mitigation solutions based on optical clamping.

Contents

Contents i

List of Acronyms iii

List of Symbols vii

List of Tables ix

List of Figures xi

1 Introduction 1

1.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Raman amplification - state of the art . . . . . . . . . . . . . . . . . . . . . 5

1.3 Objectives and thesis organization . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4.1 List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Raman amplification theory 19

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Single pump amplification . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1 Pump depletion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Performance limiting factors . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.1 Amplified spontaneous emission (ASE) . . . . . . . . . . . . . . . 29

2.4.2 Multipath interference . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4.3 Pump noise transfer . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Modeling Raman fiber amplifiers 37

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

i

List of Figures

3.3 Numerical approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.1 Shooting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.2 Collocation Method . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4.1 Lyapunov second method . . . . . . . . . . . . . . . . . . . . . . . 47

3.4.2 Second equilibrium point . . . . . . . . . . . . . . . . . . . . . . . 48

3.5 Average Power Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.6 Experimental validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.7 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Gain optimization of Raman amplifiers 59

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 Multi-pump allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3 Metaheuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4 Hybrid Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.5 Simulation implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.5.1 Dimensioning of operators . . . . . . . . . . . . . . . . . . . . . . 69

4.5.2 Dimensioning hybrid GA . . . . . . . . . . . . . . . . . . . . . . . 73

4.6 Experimental validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.7 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5 Optical amplifiers in access networks 87

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.2 Transients in Raman amplifiers . . . . . . . . . . . . . . . . . . . . . . . . 91

5.3 Raman amplification in hybrid WDM/TDM-PON . . . . . . . . . . . . . 95

5.4 Hybrid amplification schemes . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.4.1 EDFA modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.4.2 EDFA transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.4.3 EDFA transient mitigation . . . . . . . . . . . . . . . . . . . . . . . 114

5.4.4 Transmission at 10 Gb/s . . . . . . . . . . . . . . . . . . . . . . . . 127

5.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6 Conclusions 143

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.2 Directions for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

ii

List of Acronyms

Notation Description

APA Average Power Analysis.

ASE Amplifier Spontaneous Emission.

ATM Asynchronous Transfer Mode.

AWG Arrayed-Waveguide Grating.

BERT Bit Error Rate Tester.

BPON Broadband PON.

BVP Boundary Value Problem.

CapEx Capital Expenditures.

CO Central Office.

CW Continuous Wave.

CWDM Coarse Wavelength Division Multiplexing.

DCF Dispersion Compensating Fiber.

DFB Distributed Feed-Back.

DS Down-Stream.

DSF Dispersion Shifted Fiber.

DWDM Dense Wavelength Division Multiplexing.

EDF Erbium Doped Fiber.

EDFA Erbium Doped Fiber Amplifier.

EPON Ethernet PON.

FBG Fiber Bragg Grating.

FTTH/P Fiber-to-the-Home/Premise.

FWM Four-wave Mixing.

GA Genetic Algorithm.

GPON Gigabit PON.

iii

List of Acronyms


IP Internet Protocol.

IVP Initial Value Problem.

MPI Multi Path interference.

MZI Mach-Zehnder Interferometer.

ODE Ordinary Differential equation.

OLT Optical Line Terminal.

ONU Optical Network Unit.

OpEx Operational Expenditures.

OSNR Optical Signal to Noise Ratio.

PON Passive Optical Network.

PRBS Pseudo Random Bit Sequence.

QoS Quality-of-Service.

RFA Raman Fiber Amplifier.

RN Remote Node.

SARDANA Scalable Advanced Ring-based passive Dense Access

Network Architecture.

SOA Semiconductor Optical Amplifier.

SRS Stimulated Raman Scattering.

SSMF Standard Single Mode Fiber.

TDM-PON Time Division Multiplexing - Passive Optical Network.

US Up-Stream.

VOA Variable Optical Attenuator.

WDM Wavelength Division Multiplexing.

WDM-PON Wavelength Division Multiplexing - Passive Optical

Network.

WDM/TDM-PON Wavelength Division Multiplexing/Time Division

Multiplexing - Passive Optical Network.

iv

List of Acronyms


YDFA Ytterbium Doped Fiber Amplifier.

v

List of Symbols

List of Symbols

Symbol Description

h Planck Constant.

c Speed of the light in vaccum.

kB Boltzmann Constant.

PP Pump power.

Ps Data signal power.

PASE Amplified spontaneous emission power.

PSRB Single Rayleigh backscattering power.

PDRB Double Rayleigh backscattering power.

αp Fiber attenuation at the pump wavelength.

αs Fiber attenuation at the data signal wavelength.

gR Raman gain efficiency.

γR Raman gain coefficient.

TR Raman time constant.

Leff Effective length.

Gon/off On/off gain.

Gnet Net gain.

ns data signal number of photons.

np pump number of photons.

T Absolute temperature.

η(T ) Phonon occupancy factor.

NF Noise figure.

NFeff Effective noise figure.

RIN Relative intensity noise.

V Group velocity.

J Jacobi matrix.

n Refractive index.

n2 Nonlinear refractive index.

χ(3) Third order nonlinear susceptibility.

τ Fluorescence lifetime.

D Relative inversion of population.

vii

List of Symbols

Symbol Description

σ12 Absorption cross section.

σ21 Emission cross section.

S Active area.

P IS Intrinsic saturation power.

τe Characteristic decay time.

γ Saturation factor.

viii

List of Tables

3.1 Stability properties of linear systems. . . . . . . . . . . . . . . . . . . . . . 45

4.1 Optimized pumps wavelengths and powers. . . . . . . . . . . . . . . . . 70

4.2 Optimized ripples (means and standard deviations) obtained in 10 trials

at the end of the first generation for population sized with 25, 50, 75 and

100 individuals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71


for different types of selection methods at the end of 10 generations. . . 72


for different types of crossover methods at the end of 10 generations. . . 72


for different types of mutation methods at the end of 10 generations. . . 73

5.1 Tested bit sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2 Summary of possible scenarios of deployment of SARDANA network. . 97

5.3 Channel dropping order along the ring (2 channels per node) for

Situation A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.4 Channel dropping order along the ring (2 channels per node) for

Situation B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.5 Tested bit sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

ix

List of Figures

1.1 Representation of the ITU wavelength grid [4]. . . . . . . . . . . . . . . . 2

1.2 Optical signal power evolution in a distributed and in a lumped

amplification scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Number of citation in the IEEE database of optical amplifiers namely

RFA, EDFA and SOA from 1975 to the present. Lines are visual guides

for the eyes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Sheme describing inelastic spontaneous Raman Scattering (top) with

the Sokes radiation, (middle) anti-skokes radiation. (bottom) Elastic

Rayleigh radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Sheme describing stimulated Raman scattering. . . . . . . . . . . . . . . 21

2.3 Raman gain coefficient spectra for two germanosilicate fibers: Standard

Single Mode Fiber (SSMF) and Dispersion Compensating Fiber (DCF),

for a pump wavelength of 1450 nm. . . . . . . . . . . . . . . . . . . . . . 22

2.4 General scheme for a distributed Raman amplifier. For simplicity the

optical isolators used to protect the pumps and signals sources, were

omitted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Pump power evolution along 40 km of SSMF in the small signal

approximation and considering pump depletion for an input signal

power equal to 1 mW, 10 mW and 50 mW. . . . . . . . . . . . . . . . . . . 25

2.6 Signal power evolution along 40 km of SSMF in the small signal

approximation and considering pump depletion. The initial pump

power is equal to 500 mW. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.7 Pump power evolution along 40 km of SSMF in the small signal

approximation and considering pump depletion for an input signal

power equal to 1 mW, 10 mW and 50 mW. . . . . . . . . . . . . . . . . . . 26

2.8 Net gain and effective noise figure spectra for a system with two

bidirectional pumps and 25 × 400 GHz data signals along a span of

50 km SSMF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.9 Noise Figure spectra for temperatures equal to 300 K, 200 K, 100 K and

0 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

xi

List of Figures

3.1 Pump power solutions obtained through direct shooting along the fiber

length. (solid line) - Enhanced RK4, (dashed line) - Simple RK4. . . . . . 43

3.2 Phase portrait. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3 Sheme of the implemented method for one data signal and one

backward pump. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4 Raman gain efficiency spectrum. The points were obtained

experimentally according to [27]. . . . . . . . . . . . . . . . . . . . . . . . 50

3.5 Spatial evolution of 2 counter propagated pumps, 1 co propagate pump

and 4 probe signal along a 40 km SSMF fiber span amplifier. Probe

signals evolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.6 Spatial evolution of 2 counter propagated pumps, 1 co propagate pump

and 4 probe signal along a 40 km SSMF fiber span amplifier. Pump

signals evolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.7 Scheme of the experimental setup. . . . . . . . . . . . . . . . . . . . . . . 52

3.8 Measured optical spectrum for the 4-pump Raman module at 20

percent of the total power (A 3 dB attenuator was considered in this

measurement). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.9 Measured and simulated on/off gain. . . . . . . . . . . . . . . . . . . . . 53

4.1 Gain enlargement and equalization using multi-pump allocation. . . . . 61

4.2 Crossover and mutation operation scheme. . . . . . . . . . . . . . . . . . 64

4.3 The generic evolutionay algorithm. . . . . . . . . . . . . . . . . . . . . . . 64

4.4 Scheme of the Nelder-Mead algorithm. . . . . . . . . . . . . . . . . . . . 69

4.5 Scheme of the implemented setup for the 5 pumps scenario. . . . . . . . 70

4.6 Optimized ripples at the end of the GA and the hybrid GA for different

number of generations for a population size of 50 individuals. The line

is the exponential interpolation. . . . . . . . . . . . . . . . . . . . . . . . . 74

4.7 Relative Nelder-Mead time and the corresponding ripple variation. The

lines are visual guides. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.8 The GA, the Nelder-Mead and the total simulation times for a hybrid

GA against the number of generations for a population size of 50

individuals. The lines are visual guides. . . . . . . . . . . . . . . . . . . . 75

4.9 Scheme of the used experimental setup. . . . . . . . . . . . . . . . . . . . 76

4.10 Gain spectrum for the single pump scheme. Only the experimental

bandwidth is assigned in the graphs. . . . . . . . . . . . . . . . . . . . . . 77

4.11 Gain spectrum for multipump scheme. Only the experimental

bandwidth is assigned in the graphs. . . . . . . . . . . . . . . . . . . . . . 78

4.12 Measured and simulated on/off gain for optimized pumping power. . . 79

4.13 Power evolution of optimized pumps along 20 km of SMF (lines). The

geometric shapes stand for the used experimental values. . . . . . . . . . 80

xii

List of Figures

4.14 Experimental (arrows) and simulation (line) on/off spectral gain for the

20 probe signals and 7 counter propagated pumps, over 20 km of SMF

fiber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.1 General scheme of a PON and its subnetworks: feeder and distribution. 87

5.2 Two different PON architectures: (left) TDM-PON; (rigth) WDM-PON [5]. 89

5.3 Bit sequence at the receiver: top - Raman amplification, bottom- EDFA

amplification. The vertical scale is arbitrary and the horizontal scale is

500 ns/div. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.4 Experimental BER for the Raman and EDFA amplification, and the back

to back situation. The lines are visual guides. . . . . . . . . . . . . . . . . 94

5.5 Packet signals bit sequences, the time between burst is 2 µs. Packet

occupancy densities: from left to right and from top to bottom: 90, 70,

50, and 20 percent. The vertical scale is arbitrary and the horizontal

scale is 500 ns/div. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.6 Q factor as function of the packet occupancy density. The lines are

visual guides.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.7 SARDANA architecture scheme. . . . . . . . . . . . . . . . . . . . . . . . 96

5.8 Net gain and OSNR spectra after dropping for normal scenario. Lines

are guides for the eyes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.9 Net gain spectrum after dropping for resilient scenario. Lines are

guides for the eyes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.10 OSNR spectrum after dropping for the resilient scenarios. Lines are

guides for the eyes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.11 Pump power value in each node. . . . . . . . . . . . . . . . . . . . . . . . 100

5.12 Energy level diagram corresponding to a Stark split three level laser

system. The symbols A±NR indicate the thermalization between adjacent

Stark sub-levels, while W and A denote stimulated and spontaneous

emission or absorption rates, respectively and R pump rate. . . . . . . . 102

5.13 Erbium ion energy-level scheme. . . . . . . . . . . . . . . . . . . . . . . . 102

5.14 Emission and absoption cross sections spectra for alumino

germanosilicate Er3+ glass fibers. . . . . . . . . . . . . . . . . . . . . . . . 103

5.15 Gain coefficient spectra for different relative inversion of population

for an alumino-germanosilicate Er fiber. D = −1 indicates that all the

ions are in the gound state, while D = +1 denotes a fully inversion of

population. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.16 Net gain spectra at the output of three EDFA with 2.5 m , 5.0 m and

7.5 m, respectively. The pump is a 1480 nm diode laser with 100 mW of

power forwardly injected into the EDF. . . . . . . . . . . . . . . . . . . . 106

xiii

List of Figures

5.17 216−1 bit PRBS packet - 50000 bit idle obtained in the oscilloscope at the

receiver for a Raman amplification with 40 km of SSMF. The horizontal

scale is 5.000 µs/div and the vertical scale is arbitrary. . . . . . . . . . . . 110

5.18 Visualized packets and their envelope exponential fit. . . . . . . . . . . . 112

5.19 Packet decaying time as a function of the idle time. The line represents

an exponential fit in the form A0 exp (B0) + C0. A0 = 3.32 ± 0.27 µs,

B0 = 36.39± 10.22 µs, C0 = 1.35± 0.25 µs, r2 = 0.9803. . . . . . . . . . . 113

5.20 Visualized sequences at the receiver: (a) Sequence C, (b) Sequence D.

Horizontal scale 20.00 µs/div and vertical scale is arbitrary. . . . . . . . . 114

5.21 Visualized sequences at the receiver: (a) Sequence A, (b) Sequence B.

Horizontal scale 5.000 µs/div and vertical scale is arbitrary. . . . . . . . . 114

5.22 Q factor as a function of the occupancy density. The lines are visual

guides. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.23 Scheme of the experimental setup used for the gain clamping. . . . . . . 116

5.24 Spectra captured by the OSA. Horizontal scale 0.47 nm/div, vertical

scale 10 dB/div. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.25 Spectra captured by the OSA. Horizontal scale 0.47 nm/div, vertical

scale 10 dB/div. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.26 (left) Packets gain excursion as a function of the re-injected signal

power. (right) System signal-to-noise ratio as a function of the re-

injected signal power with traffic based on packets and PRBS of length

231 − 1 and 215 − 1. The receiver power is kept constant equal to -8 dBm. 119

5.27 Gain excursion as a function of the feedback signal power for packets

with idle time equal to 5 µs (50000 bits), 10 µs (100000 bits) and 20 µs

(200000 bits) at 10 Gb/s. The receiver power is kept constant equal to

-8 dBm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.28 Signal-to-noise ratio as a function of the feedback signal power for

packets with idle time equal to 5 µs (50000 bits), 10 µs (100000 bits)

and 20 µs (200000 bits) at 10 Gb/s. The receiver power is kept constant

equal to -8 dBm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.29 Generic clamping scheme feasible in a WDM/TDM-PON remote node

(RN) in which two channels are dropped / added. . . . . . . . . . . . . . 122

5.30 Packet gain excursion as function of control signal power for an EDF

with 15 m. (black squares) fForward clamping, (red circles) backward

clamping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.31 Scheme of the implemented experimental setup. . . . . . . . . . . . . . . 123

5.32 Back-to-back. (Left)-Pattern- horizontal scale 10.00 µs/div, vertical scale

30 mV/div. (Right) eye diagram- horizontal scale 200 ps/div, vertical

scale 30 mV/div. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

xiv

List of Figures

5.33 Receiver output with the CW signal off. (Left)-Pattern- horizontal scale

10.00 µs/div, vertical scale 100 mV/div. (Right) eye diagram- horizontal

scale 100 ps/div, vertical scale 100 mV/div. . . . . . . . . . . . . . . . . . 124

5.34 Packet gain excursion as function of control signal power. Continuous

lines are linear fits. (squares) - Simple clamping, adjusted r square=

0.79; (circles) enhanced clamping, adjusted r square = 0.73. The dashed

line stands for the gain excursion level for the unclamped situation. . . . 125

5.35 Q factor at the receiver as a function of the clamping control signal

power. (squares) - Simple clamping and (circles) enhanced clamping.

The dashed line stands for the Q-factor level for unclamped situation. . 126

5.36 Scheme of the implemented experimental setup. . . . . . . . . . . . . . . 127

5.37 Optical spectrum at the output of (a) the negative dispersion fiber and

(b) the receiver. The vertical and horizontal scales are 10 dB/div and

2.00 nm/div, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.38 Eye diagram at the receiver when all the low frequency modulated

channels are (left) off and (rigth) on. . . . . . . . . . . . . . . . . . . . . . 129

5.39 Signal pattern, of the test signal, in the presence of add/drop function.

The vertical scales are arbitrary. . . . . . . . . . . . . . . . . . . . . . . . . 129

5.40 BER versus the optical power at the receiver. The squares represent the

back-to-back, circles represent all the channels on and triangle shown

the situation where only channels X is on. Lines are guides for the eyes. 130

5.41 Pattern and eye diagram at the end of the: (a) Tx, (b) Rx with fiber 1 and

(c) Rx with fiber 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.42 BER curves for back to back, fiber 1 and fiber as a function of the power

at receiver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.43 Raman gain efficiency for pumping at 1480 nm. The channels are

represented by arrows (black-C band and red-L band). The curve was

obtained by interpolation of experimental data [57]. . . . . . . . . . . . . 133

5.44 (Top)- Net gain dropping spectrum after dropping. Squares- simple

Raman. Circles- optimized hybrid Raman/ in line EDFA. The

horizontal dashed line settles a detection threshold. (Middle)-

Optimized spans of EDF and their positions along the ring. The vertical

dashed lines are used to place the dropped channel wavelength along

the ring. The total pump power is equal to 1 W. (Bottom) - Available

power to pump the EDF along the ring. . . . . . . . . . . . . . . . . . . . 134

xv

Chapter 1

Introduction

1.1 Background and motivation

The unprecedented popularity of Internet and its derived new services such as

video streaming delivery (HDTV, IPTV, etc) and file sharing applications (Peer-to-Peer

for example) have pressed out the competition among providers in order to reduce the

cost of the transmission systems.

In the 90s, the use of optical amplifiers brought a great development in the

simplification of those systems by eliminating expensive and limitative optoelectronic

regeneration. Thus substantial improvements have been brougth to transmission

systems in terms of cost, flexibility design and capacity. Before them, the standard

method to compensate the power loss in optical fiber was to use periodically spaced

electronic regenerators along the transmission link.

In those expensive devices, the signal in the optical domain was converted to the

electric domain using a photodetector, being processed, amplified and then reinjected

into the optical fiber by a transmitter. One advantage of this technology is that

the transmission impairments such as noise, dispersion and nonlinearities do not

accumulate along the link, being compensated in each node. However, system

upgrades, such as bit rates and modulation formats are expensive and difficult to

implement because all the regenerators of the link have to be replaced. Besides that,

the huge bandwidth of the optical fiber cannot be exploited properly.

The introduction of wavelength division multiplexing (WDM) is of great

importance because it enabled the implementation of the actual high capacity optical

fiber networks by fully exploiting the fiber bandwidth [1]. WDM exists in several

frequency patterns which differ in the spacing of the wavelengths, number of

channels, and the optical amplification ability. Conventional WDM systems provide

up to 16 channels in the 3rd transmission window (C-Band) of Silica fibers around

1550 nm, while dense wavelength division multiplexing (DWDM) uses the same

transmission window but with denser channel spacing. Channel plans vary, but a

typical system would use 40 channels at 100 GHz spacing or 80 channels with 50 GHz

spacing. Some technologies are capable of 25 GHz spacing (sometimes called ultra

dense WDM). These solutions are extensively used in long haul systems, sending

as much information as possible. New amplification options (Raman amplification)

1. Introduction

enable the extension of the usable wavelengths to the L-band by doubling these

numbers [2]. Coarse wavelength division multiplexing (CWDM) uses increased

channel spacing to reduce the network cost by the use of less sophisticated transceiver

designs. Thus, to provide 16 channels on a standard single mode fiber (SSMF), CWDM

uses the entire frequency band between the second and third transmission window

(1310/1550 nm respectively) including both windows (minimum dispersion window

and minimum attenuation window) but also the critical region where OH absorption

may occur [3]. A representation of the ITU wavelength grid is depicted in Figure 1.1.

Figure 1.1: Representation of the ITU wavelength grid [4].

CWDM is a good solution whenever less information is transmitted over short

distances in a less expensive way than DWDM. Their progressive introduction

had become possible with the development of optical amplification, in particular,

Erbium doped fiber amplifier (EDFA) which have revolutionized the field of optical

communications due to the high deployed gain with relatively low power optical

pumps. However, as the channels in CWDM systems are far apart, optical

amplification is still a matter of concern because the EDFAs bandwidth (20 − 40 nm)

cannot support the full band of CWDM channels [5].

Nevertheless, the EDFAs had undergown progressive improvements, and their

initial bandwidth was enlarged from the C band to the L band, increasing the

capacity of WDM systems. Besides that, new rare-earth fiber amplifiers had been

implemented, namely Ytterbium doped fiber amplifier (YDFA), which presents high

absorption/emission bands (from 850 nm to 1200 nm) [6]. Ytterbium can also be

used together to Erbium as a co-dopant increasing the absorption because they can

be pumped eith by the core or by the cladding [7], [8].

However, the EDFA bandwidth and its co-doped derivants was still must lower

2

1.1. Background and motivation

than the transmission band of low loss fibers. Furthermore, EDFAs also have other

disadvantages, such as the use of a lumped amplification scheme. In this scheme,

a span of optical fiber is followed by a span of pumped Erbium doped fiber (EDF) at

every 40-50 km. This distance is determined by the span loss and by the limit imposed

from the maximum admissible power allowed in the fiber, without inducing nonlinear

effects and the minimum acceptable power that avoids a large degradation of the

optical signal-to-noise ratio (OSNR). The use of distributed amplification scheme

could increase the transmission distance because it allows the the contention of the

signal inside the limits imposed by the nonlinearities and of the signal-to-noise-ratio

degradation resulting from higher span distances. This advantage of the distributed

(Raman) over lumped amplification in containing the signal power is illustrated in

Figure 1.2. The distributed amplification scheme can be used to cover very long span

links or to increase the distance of ultra-long haul systems.

SignalPower

Distance

EDFA EDFA EDFA

High nonlinearities

High noise

Distributed

Lumped

Figure 1.2: Optical signal power evolution in a distributed and in a lumped amplificationscheme.

As a consequence, the interest on broad-band distributed amplifiers has been

increasing, as depicted in Figure 1.3, where a number of optical amplifiers, (EDFA,

Raman and SOA) citations in the IEEE database is plotted from 1975 to the present

[9]. On looking at this plot, we notice that there is a research interest on Raman

amplification since the 80s of the XX century. However, with the emergence of the

EDFA in the 90s, the attention given to Raman amplification had decreased in that

period. In the begging of the XXI century, the interest in Raman amplification was

renewed, in part because the high power pump become available commercially at

a reasonable cost and also because the bandwidth of the EDFA was already fully

utilized, enabling the need for more optical channels and wider optical bandwidth.

The research interest in semiconductor optical amplifier (SOA) has also augmented

3

1. Introduction

in last years. Given their relatively broad gain bandwidth, and applicability to all

optical communication bands from 1300 through 1600 nm and beyond, SOAs are

natural candidates for applications requiring a versatile, inexpensive broad-band

amplifier [10], [11]. SOA are also becoming noticed due to other applications, such

as optical switchers in optical packet networks [12] or integrated with Mach-Zehnder

interferometer (MZI) to provide optical logic XOR gate for several kinds of all-optical

device, such as comparators, adders and counters [13].

1975 1980 1985 1990 1995 2000 2005 2010

1

10

100

IEE

E E

xplo

rer C

itatio

ns

Year

Raman EDFA SOA

Figure 1.3: Number of citation in the IEEE database of optical amplifiers namely RFA, EDFAand SOA from 1975 to the present. Lines are visual guides for the eyes.

One possible solution for a near future amplification solution could rely on Raman

fiber amplifiers, because they present flexible transmission band given the pumps

frequencies. This behaviour is actually different from the rare-earth doped fiber

amplifiers that depend on the energy bands of the dopants. Another benefit is that the

transmission medium is also the gain medium (the amplification scheme is distributed

not lumped). Hence, it allows the contention of the signal inside the limits imposed by

the nonlinearities and by the OSNR degradation resulting from higher span distances.

It must be stessed out that the distributed amplification scheme can be used to cover

very long span links or to increase the distance of ultra-long haul systems. Besides the

above mentioned merits, others advantages can be accounted for [14], [15], [16]:

• the gain depends only on the frequency separation between pump and signal

and not absolute frequencies. Therefore, it is possible to obtain gain at

any wavelength by choosing properly the pump frequency. The addition of

4

1.2. Raman amplification - state of the art

more pumping lasers enlarges the gain bandwidth necessary for broadband

applications.

• the gain does not depend on the relative direction of propagation of a pump and

signal.

• the noise performance is very good when compared to EDFA.

• the gain transience with burst traffic is negligible.

Besides that, Raman amplifiers have the merit of arbitrary gain band. The technique

to shape the gain profile over a wide bandwidth is quite simple because it deals only

with the optimization of pumping parameters, initial powers and wavelengths.

High bitrate optical networks with burst traffic are expected to be the next

generation of photonic transmission systems. Optical burst switching (OBS) has

been proposed as a technique to overcome issues related to WDM deployment, like

lack of fine granularity in wavelength routing and electronic speed bottleneck in

SONET/SDH. In these networks the traffic is, in a certain scale, uniform, due to the

use of the protocol coding, which restricts the number of adjacent zero binary bits.

However, the future optical networks can have packet traffic directly over the optical

channels, imposing a lack of optical signal for a large period of time. This will lead to

possible gain transients on the optical amplifiers. In fact one of the constrains for

burst traffic in fiber networks arises from optical amplification, where the Erbium

Dopped fiber amplifiers (EDFAs) present some limitations, ensuing from their long

carrier lifetime [17]. The key point is, in fact, to enhance WDM technologies so that

higher bit-rates per channel OBS traffic can be transmitted, moving them from the

transport segment up to the Metro and Access segment [10]. It was unavoidable that

the pursuit of such improvements in WDM technologies would narrow the margins

of system design. For this reason, researchers have begun seeking alternative enabling

technologies capable of mitigating system impairments, such as noise, transients, non-

linear effects and handling wider bandwidths [1]. Raman amplification can be pointed

out as a potential solution for the amplification of burst optical networks. Potentially,

Raman amplifiers can overcome the EDFAs performance.

1.2 Raman amplification - state of the art

The fundamentals of Raman scattering was established on 1928 [18] but their

application as optical amplifiers was only proposed in the 70s and 80s of the

XX century [19], [20], with the discovery of stimulated Raman scattering (SRS) in

1962 [21]. Yet, since those initial amplifiers were not very competitive due to the

5

1. Introduction

cost and lack of reliability of the needed high power pumping lasers, they had

remained underestimated, especially after the invention of the EDFA. However,

recent developments in the field of high power lasers had enabled the emergence of

affordable devices and as a consequence, the interest on Raman fiber amplifier (RFA)

was renewed [22].

Raman fiber amplifiers are based on the power transfer from pump(s) signal(s) to

information carrying signals (usually described as probes) due to SRS which occurs

when there is sufficient pump power within the fiber. Since the gain peak of this

amplification is obtained for signals downshifted approximately by 13.2 THz (for

Silica), relative to the pump signal frequency, to achieve gain at any wavelength

we need to select a pump signal whose frequency complies with this relation. In

this way, it is possible to optimize the number of pumps to obtain a wide and flat

gain [23], [24], [25]. However, it is necessary to bear in mind that due to the pump-to-

pump interaction, the shorter wavelength pumps demand more power to be effective

[26], [27].

From a telecommunications point of view, the pump signals wavelengths must be

placed around 1450 nm because the probe signal wavelengths used on the so called

3rd transmission window are centered around 1550 nm and the maximum gain occurs

for a Stokes frequency shift of 13.2 THz. Typically a high power laser for Raman

amplification, provide an optical power of 300 mW, launched over an optical fiber,

which for a SSMF is equivalent to a power density of 3.75 GW/m2. This high power

injected in to the fiber, especially when multi-pump lasers are utilized, imposes new

concerns in terms of safety. One of the problems arising from the high powers is the

fiber fuse effect, named due to the similarity with a burning fuse. This phenomenon

can lead to the destruction of the optical fibers along kilometers. The fiber fuse is

initiated by a local heating of an optical fiber, causing a strong light absorption that

increases the temperature up to the Silica vaporization value. In this high temperature

region the core of the fiber vaporizes emitting a visible radiation. Due to the heating

transmitted to the neighboring regions the process propagates towards the high-

power light sources [28], [29].

In terms of implemented systems, several architectures have been proposed,

based in all Raman or hybrid Raman/EDFA amplification [30]. The use of

bidirectional Raman amplification has also been reported for long reach access

networks. Experimental results have shown the feasibility of systems with symmetric

up-and-downstream signals with bitrates up to 10 Gb/s, supported by distributed

Raman amplification over 80 km of fiber [31]. Field transmission experiments have

been reported with 8× 170 Gb/s over 210 km of single mode standard fiber, achieving

spectral efficiency of 0.53 bit/s/Hz [32].

As said previously, RFA provide the feasibility of wide and flat spectral gain

profiles with the combination of several pumping lasers operating at specific powers

6

1.2. Raman amplification - state of the art

and wavelengths. The composite amplification is determined from the mutual

interactions among the pump and signal wavelengths. Gain spectra as large as 100 nm

were obtained using multiple pumps. Emori et al [33] have presented an experimental

Raman amplifier with a 100 nm bandwidth using a WDM laser diode unit with 12

wavelengths ranging from 1405 to 1510 nm, whose maximum total power was equal

to 2.2 W. By this way, a gain equal to 2 dB is obtained over a 25 km SSMF link and

a 6.5 dB gain using a 25 km dispersion shifted fiber (DSF) link, both with 0.5 dB of

maximum ripple. Kidorf et al [26] provided a mathematical model to implement a 100

nm Raman amplifier using low power pumps with maximum power of each pump

equal to 130 mW. They used 8 pumps from 1416 nm to 1502 nm along 45 km of SSMF,

obtaining a gain around 4 dB with a maximum ripple equal to 1.1 dB.

Enlargement of the bandwidth of Raman amplifiers is also achieved using

incoherent pumping instead of multi-pump schemes [34], [35], [36]. Vakhshoori et

al proposed a high-power incoherent semiconductor pump prototype, that use a low-

power seed optical signal, coupled into a long-cavity semiconductor amplifier, has

achieved 400 mW of optical power over a 35 nm spectral window [37]. A 50 nm

bandwidth amplifier was obtained with an on/off gain equal to 7 dB. It was also

demonstrated that the use of six coherent pumps is less efficient, in terms of flatness,

than the use of two incoherent pumps [4]. The signal wavelengths were comprised

between 1530 nm and 1605 nm and the transmission occurs over 100 km of optical

fiber. Another advantage of using incoherent pumping is the reducing of nonlinear

effects, such as Brillouin scattering, four-wave mixing (FWM) induced by pump-

pump, pump-signal and pump-noise interactions [38].

The growing maturity of high pump module technologies is providing competitive

solutions based on Raman amplification and currently many alternative techniques

are being developed to overcome the ordinary one pump and dual pumping methods

[39]. In particular, two major techniques are highlighted. First, the use of low

power pumping lasers provides gain comparable to the ordinary one pump Raman

amplification. This technique is especially interesting for combining commercial and

low cost lasers [40]. The second particular technique corresponds to an evolution of

the cascaded Raman amplification. Actually, a sixth order cascade Raman amplifier

was recently proposed [41]. In the cascade Raman amplification, the pump power

is downshifted in frequency by using a pair of fiber Bragg gratings (FBGs) placed in

spectral positions multiples of 13 THz, from the pump frequency. In a particular case,

the generation of the fiber pump laser is obtained by using only one passive reflector

element and distributed reflectors over the long optical fiber, establish by a nonlinear

fiber intrinsic effect called Rayleigh backscattering.

7

1. Introduction

1.3 Objectives and thesis organization

The goal of this thesis is to study the RFA in the scope of modern optical

communication systems, accounting for their principal challenges [42]. Specific topics

such as, the development of an efficient simulator for Raman amplified links, the

gain enlargement and equalization, hybrid Raman/EDFA schemes and the effect of

transients are important and thus addressed both theoretically and experimentally.

The work reported here was mainly performed in the facilities of the Instituto

de Telecomunicações (IT) in Aveiro and had the advantage of being framed by

both portuguese (TECLAR, ARPA, FEFOF, TOMAR-PON) and European projects

(BONE, SARDANA and EURO-Fos). A short period training in the National Intitute

of Information and Communications Technology (NICT) sitted in Tokyo, Japan,

supervised by Professor Nayoa Wada, enabled the access to top laboratorial facilities

and the accompliment of important work.

This thesis is organized in six chapters, in which several aspects of Raman

amplification in telecommunications are studied.

Chapter 1 provides a background and a motivation for this work and an overview

of Raman amplification within the frame of optical communications systems. Raman

amplification principles of operation are introduced, being their pros and cons

discussed. The most relevant challenges relating this topic are presented as well as

state of the art work. Chapter 1 also presents a summary of results provided by a list

of published work.

Chapter 2 presents the Raman amplification theory, namely SRS. A mathematical

model for propagation in optical fibers is then introduced and the relevance of

pumping shemes in optical communications analyzed. Since the computation of

solutions is a cumbersone task, a relevant approximation, small-signal approximation,

is presented, and its most important results discussed. The effect outside the

approximation, pump depletion, is also discussed. An analysis of the noise behaviour

of RFA is also presented, being their implications for communications systems

discussed.

Chapter 3 is focused on modeling of RFA. Hence, a general mathematical model

for multi-pump, multi-signal given by a system of nonlinear ordinary differential

equation (ODE) is presented. Possible methodologies to solve the equations are

presented and discussed. The latter, can be either numerical (shooting, collocation,

etc) or semi-analytical. The use of average power analysis (APA) is introduced and

their improvements in terms of accuracy and efficiency discussed. Other qualitative

analytical approaches, such as the linearization in the vicinity of critical points or

Lyapunov-second methods are also analyzed and applied to the problem of RFA.

Chapter 4 is devoted to the topic of gain enlargement and optimization by

allocation of pumps, which is a very important application of RFA. A method based

8

1.4. Summary of results

on an hybrid Genetic Algorithm is introduced. This method uses a combination of an

heuristic, GA, with a local search, Nelder-Mead. It is demonstrated that the proper

configuration of this method allows a improvement of a factor of 2 in its efficiency.

An experimental validation of the gain optimization and gain enlargement is also

provided.

Chapter 5 regards the practical application of optical amplifiers in access networks.

The work presented is mainly focused on amplifiers transients, RFA and EDFA, in

the scope of optical bursty/packeted traffic. The latter were demonstrated to be

more penalizing for EDFA, and experimental characterizations were carried on to

assess the decays times. Two mitigation solutions based on optical gain clamping are

proposed, one based on a feed-back ring and another based on enhanced bidirectional

clamping with FBG. Their efficiency for WDM/TDM-PON is also discussed. This

chapter also addresses the topic of hybrid Raman/EDFA amplification, namely for

C+L band. Some practical situation are assessed, namely WDM ring in rural scenarios.

The feasibility of this amplification schemes for normal and resilient modes is also

discussed.

This thesis ends with chapter 6 where the main conclusions are summarized and

directions for future work are pointed out.

1.4 Summary of results

The most important results accomplished from this thesis are:

• The development of an efficient and accurate simulator for Raman fiber

amplifiers using average power analysis and its experimental validation .

• The implementation of a pump allocation algorithm for gain equalization using a

combination of genetic algorithm and Nelder Mead method and its experimental

validation.

• The assessment of extended and equalized gain in C+L transmission bands using

hybrid Raman/EDFA for WDM/TDM PON solutions in rural scenarios.

• The assessment and charaterization of transients, Raman and EDFA and the

combination of both in networks with bursty traffic.

• The implementation of mitigation solutions for EDFA transients based on optical

gain clamping, suitable for metro and access networks.

• The evaluation of the above mentioned effect of transmission systems at 10 Gb/s.

The work accomplished within the period of this thesis resulted in 10 published

papers in journals and another under peer revision, 26 conference proceedings and

9

1. Introduction

also a chapter in a book. The work has also originated colaborations within other

researchers in the optical communications group, which were also published. The

publication are listed bellow .

1.4.1 List of publications

Chapters in books

1. J. Schlesinger, Optical Fibers Research Advances. Nova Science Pub Inc, 2007.

Papers in journals

1. B. Neto, A. Teixeira, N. Wada, and P. André, “Efficient use of hybrid genetic

algorithms in the gain optimization of distributed raman amplifiers,” Optics

Express, vol. 15, no. 26, pp. 17 520–17 528, 2007.

2. D. Sperti, P. André, B. Neto, A. Rocha, A. Bononi, F. da Rocha, and

M. Facão, “Experimental assessment of some Raman fiber amplifiers solutions

for coarse wavelength division multiplexing applications,” Photonic Network

Communications, vol. 16, no. 3, pp. 195–202, 2008.

3. A. Rocha, B. Neto, M. Facao, and P. Andre, “Study of raman amplification with

low cost incoherent pumps,” Microwave and Optical Technology Letters, vol. 50,

no. 2, pp. 301–303, 2008.

4. P. André, B. Neto, A. Teixeira, and N. Wada, “Raman amplification impact in

packet base networks,” Microwave and Optical Technology Letters, vol. 50, no. 12,

pp. 3083–3085, 2008.

5. A. Rocha, B. Neto, M. Facao, and P. Andre, “Low cost incoherent pump solution

for Raman fiber amplifier,” Optica Applicata, vol. 39, no. 2, pp. 287–293, 2009.

6. J. Reis, B. Neto, P. André, and A. Teixeira, “WDM ring performance

improvement by means of four-wave mixing crosstalk minimization algorithm,”

Microwave and Optical Technology Letters, vol. 51, no. 8, pp. 1949–1952, 2009.

7. B. Neto, J. Ferreira, N. Wada, A. Pinto, and P. André, “Evaluation of the

effect of channel add/drop impact on power transients on the performance

of a 10-GB/S DWDM transmission system with hybrid EDFA/Raman

amplification,”Microwave and Optical Technology Letters, vol. 52, no. 6, pp.1225–

1228, 2010.

8. M. T. M. R. Giraldi, A. M. Rocha, B. Neto, C. Correira, M. E. V. Segatto,

M. J. Pontes, A. P. L. Barbero, J. C. W. Costa, M. A. G. Martinez, O. Frazão,

10


J. M. Baptista, H. M. Salgado, B. M. Marques, A. L. J. Teixeira, P. S. André,

“Rayleigh assisted Brillouin effects in distributed Raman amplifiers under

saturated conditions at 40 Gb/s,Microwave and Optical Technology Letters, vol. 52,

no. 6, pp.1331–1335, 2010.

9. C. Reis, A. Maziotis, C. Kouloumentas, C. Stamatiadis, M. Bougioukos,

N. Calabretta, P. Andre, R. Dionisio, B. Neto, H. Dorren, H. Avramopoulos, and

A. Teixeira, “All optical synchronous SR flip-flop based on active interferometric

devices”, Electronics Letters, vol. 46, no. 10, pp. 709 –710, May 2010.

10. B. Neto, C. Reis, R. P. Dionísio, J. M. Ferreira, J. A. Lazaro, G. Tosi-Beleffi,

A. N. Pinto, R. Nogueira, A. Teixeira, J. Prat, P. S. André “Assessment and

mitigation of EDFA gain transients in hybrid WDM/TDM PON in the presence

of packet based traffic, Optoelectronics, IET, to be published, 2010.

11. B. Neto, A. Klingler, C. Reis, R. P. Dionísio, R. N. Nogueira, A. Teixeira,

P. S. André “Enhanced optical gain clamping for upstream packet based traffic

on hybrid WDM/TDM-PON using Fiber Bragg Grating, Journal of Optical

Communications and Networking, OSA, submitted, July 2010.

Papers in conference proceedings

1. P. Andre, A. Pinto, A. Teixeira, B. Neto, S. Junior, D. Sperti, F. da Rocha,

M. Bernardo, M. Fujiwara, A. Rocha, and M. Facao, “Raman amplification using

incoherent pump sources,” in Transparent Optical Networks, 2007. ICTON ’07. 9th

International Conference on, vol. 1, July 2007, pp. 136–139.

2. P. André, A. Rocha, B. Neto, and M. Facão, “Improvement of Raman

Amplification Gain Tilt Using Incoherent Pump Sources,” Laser Science, 2007.

3. A. Teixeira, P. Teixeira, B. Neto, R. Nogueira, and P. Andre, “Automatic

apodization profiling of super structured fiber Bragg gratings for OCDMA

coding applications,” in Optical Fiber Conference OFC’08, February 2008, pp.

JThA29.

4. P. Teixeira, B. Neto, R. Nogueira, P. Andre, and A. Teixeira, “Code cardinality

maximization using highly reflective SSFBG with optimum apodization

profiles,” in 10th Anniversary International Conference on Transparent Optical

Networks, 2008. ICTON 2008, 2008, pp. 55–57.

5. C. Reis, A. M. Rocha, B. Neto, N. Wada, P. Andre, “Raman amplification in high

10 Gbit/s and 40 Gbit/s packet optical networks,” in Mediterranean Winter, 2008.

ICTON-MW 2008. 2nd ICTON., 11-13, pp. 1 –4.

11

1. Introduction

6. J. Andrade, B. Neto, A. Rocha, C. Reis, A. Teixeira, and P. Andre, “Raman

amplification in the context of next-generation passive optical networks,” in

Mediterranean Winter, 2008. ICTON-MW 2008. 2nd ICTON, 11-13 2008, pp. 1

–4.

7. J. Reis, B. Neto, P. Andre, and A. Teixeira, “WDM ring performance

improvement by means of a nonlinear effects crosstalk minimization algorithm,”

in Optical Fiber Communication - incudes post deadline papers, 2009. OFC 2009.

Conference on, 22-26 2009, pp. 1 –3.

8. A. Rocha, B. Neto, A. Martins, G. Incerti, D. Forin, G. Bellefi, M. Facao, J. Pinto,

A. Teixeira, R. Nogueira, M. Lima, and P. Andre, “The effect of high power

propagation in bended fibers,” in Telecommunications, 2009. ConTEL 2009. 10th

International Conference on, 8-10 2009, pp. 303 –304.

9. P. Coelho, J. Reis, B. Neto, P. Andre, and A. Teixeira, “Monitoring of fiber

nonlinearities in wdm based passive optical networks,” in Telecommunications,

2009. ConTEL 2009. 10th International Conference on, 8-10 2009, pp. 277 –281.

10. P. Andre, B. Neto, C. Reis, A. Rocha, N. Wada, G. Beleffi, and A. Teixeira, “Raman

amplification challenges for next generation networks,” in Transparent Optical

Networks, 2009. ICTON ’09. 11th International Conference on, june 2009, pp. 1 –2.

11. C. Reis, B. Neto, R. Dionisio, G. Incerti, G. Tosi-Beleffi, D. Forin, A. Rocha,

A. Teixeira, and P. Andre, “Transience analysis of bursty traffic with erbium

doped fiber amplifiers,” in Transparent Optical Networks, 2009. ICTON ’09. 11th

International Conference on, june 2009, pp. 1 –3.

12. J. Reis, B. Neto, P. Andre, and A. Teixeira, “Optimization of passive optical

networks by means of fiber nonlinearities interference reduction,” in Transparent

Optical Networks, 2009. ICTON ’09. 11th International Conference on, june 2009, pp.

1 –4.

13. B. Neto, M. Rodrigues, E. Rocha, and P. Andre, “Stability analysis of raman

propagation equations,” in Transparent Optical Networks, 2009. ICTON ’09. 11th


14. B. Neto, A. Klingler, C. Reis, J. Girao, A. Teixeira, and P. Andre, “EDFA transient

assessment for bursty traffic,” in ICTON Mediterranean Winter Conference, 2009.

ICTON MW 2009. 3rd, 10-12 2009, pp. 1 –4.

15. B. Neto, C. Reis, A. Teixeira, P. André,

and N. Wada, “Gain equalization technique for raman amplification systems

based on the hybrid optimization algorithm,” in Microwave and Optoelectronics

Conference (IMOC), 2009 SBMO/IEEE MTT-S International, 3-6 2009, pp. 687 –689.

12


16. M. Rocco Giraldi, A. M. Rocha, B. Neto, C. Correia, M. Segatto, M. Pontes,

A. Barbero, J. Costa, M. Martinez, O. Frazao, J. Baptista, H. Salgado, M. Marques,

A. Teixeira, and P. Andre, “Brillouin effects in distributed raman amplifiers

under saturated conditions,” in Microwave and Optoelectronics International

Conference (IMOC), 2009 SBMO/IEEE MTT-S, 3-6 2009, pp. 841 –845.

17. N. Pavlovic, A. Baptista, B. Neto, A. Rocha, P. Andre, D. Form, G. Beleffi,

J. Lazaro, J. Prat, and A. Teixeira, “Demonstration of improved osnr in

ring-based pons with remotely pumped amplification,” in OptoElectronics and

Communications Conference, 2009. OECC 2009. 14th, 13-17 2009, pp. 1 –2.

18. P. S. Andre, A. M. Rocha, B. Neto, A. Martins, M. Facao, J. L. Pinto,

A. L. J. Teixeira, R. N. Nogueira, M. J. Lima, G. Incerti, D. Forin,

G. T. Beleffi“Optical fiber bending limits for optical fiber infraestructures,” in

AFRICON, 2009. AFRICON ’09., 23-25, pp. 1 –3.

19. B. Neto, C. Reis, A. Rocha, J. P. Girão, R. P. Dionísio, , S. Chatzi, F. Bonada,

J. Lazaro, J. A. and, and A. L. J. Teixeira, “Impact of transient response of optical

amplifiers operating with burst traffic,” in 7th Conference on Telecommunications,

2009. Conftele ’09., vol. 1, May 2009, pp. —.

20. A. Martins, A. M. Rocha, B. Neto, A. Teixeira, M. Facão, R. N. Nogueira, M. Lima,

P. S. André, “Modeling of Bend Losses in Single-Mode Optical Fibers,” in 7th

Conference on Telecommunications, 2009. Conftele ’09., vol. 1, May 2009, pp.

21. B. Neto, C. Reis, A. Rocha, A. L. J. Teixeira, N. Wada, and P. S. André, “Transience

response of traffic based on optical packets with optical amplifiers,” in Networks

and Optical Communications, 2009. NOC ’09. 14th European Conference on, vol. 1,

June 2009, pp. 545–550.

22. N. B. Pavlovic, A. Baptista, B. Neto, A. Rocha, D. Forin, J. A. Lazaro, J. Prat, and

A. L. J. Teixeira, “Joint raman and remote add/drop amplification scheme for

next generation pons,” in Networks and Optical Communications, 2009. NOC ’09.

14th European Conference on, vol. 1, June 2009, pp. 435–442.

23. C. Reis, R. Dionisio, B. Neto, A. Teixeira, and P. Andre, “All-optical XOR based

on integrated MZI-SOA with co-and counter-propagation scheme,” in ICTON

Mediterranean Winter Conference, 2009. ICTON-MW 2009. 3rd. IEEE, 2010, pp.

1–4.

24. B. Neto, R. P. Dionísio, A. M. Rocha, C. Reis, S. Chatzi, F. Bonada, J. A. Lazaro,

A. L. J. Teixeira, and P. S. André, “C+L band extended reach next generation

access networks through raman amplification: assessment in rural scenario,” in

13

1. Introduction

OptoElectronics and Communications Conference, 2010. OECC 2010. 15th, 5-9 2010,

pp. 1 –2.

25. B. Neto, A. Rocha, J. P. Girão, R. P. Dionísio, C. Reis, S. Chatzi, F. Bonada,

J. A. Lazaro, A. L. J. Teixeira, and P. S. André, “C+L band gain equalization for

extended reach wdm-ring pon using hybrid raman/in line edfa amplification,”

in Transparent Optical Networks, 2010. ICTON ’10. 12th International Conference on,

vol. 1, June 2010, pp. 136–139.

26. B. Neto, A. Rocha, J. P. Girão, R. P. Dionísio, C. Reis, S. Chatzi, F. Bonada,

J. Lazaro, J. A. and, and A. L. J. Teixeira, and P. S. André, “Comparative analysis

of hybrid in line edfa/raman with simple Raman amplification in WDM ring

pon for C+L band,” in Networks and Optical Communications, 2010. NOC ’10. 15th

European Conference on, vol. 1, June 2010, pp. —.

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[27] J. Bromage, “Raman amplification for fiber communications systems,” Journal of

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A. Teixeira, R. Nogueira, M. Lima, and P. Andre, “The effect of high power

propagation in bended fibers,” in Telecommunications, 2009. ConTEL 2009. 10th

International Conference on, 8-10 2009, pp. 303 –304.

[29] P. Andre, M. Facao, A. Rocha, P. Antunes, and A. Martins, “Evaluation of the

fuse effect propagation in networks infrastructures with different types of fibers,”

in Optical Fiber Communication (OFC), collocated National Fiber Optic Engineers

Conference, 2010 Conference on (OFC/NFOEC), 21-25 2010, pp. 1 –3.

[30] D. Chen, T. Xia, G. Wellbrock, J. Peterson, D.L., S. Park, E. Thoen, C. Burton,

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line side, oc-768c client side field trial using hybrid raman/edfa amplifiers,” in

Optical Communication, 2005. ECOC 2005. 31st European Conference on, 25-29 2005,

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[32] M. Schneiders, S. Vorbeck, R. Leppla, E. Lach, M. Schmidt, S. Papernyi, and

K. Sanapi, “Field transmission of 8 times;170 gbit/s over high loss ssmf link using

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[35] T. Zhang, X. Zhang, and G. Zhang, “Distributed fiber Raman amplifiers with

incoherent pumping,” IEEE Photonics Technology Letters, vol. 17, no. 6, pp. 1175–

1177, 2005.

[36] S. Wen, “Design of the pump power spectrum for the distributed fiber Raman

amplifiers using incoherent pumping,” Optics Express, vol. 14, no. 9, pp. 3752–

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its reduction in a distributed Raman fiber amplifier,” IEEE Photonics Technology

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[41] S. Papernyi, V. Ivanov, Y. Koyano, and H. Yamamoto, “Sixth-order cascaded

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17

http://link.aip.org/link/?ELL/35/1355/1


http://link.aip.org/link/?ELL/36/1356/1

Chapter 2

Raman amplification theory

2.1 Introduction

To understand the basics behind Raman amplification, it is necessary to

understand the phenomenon that describes it, stimulated Raman scattering (SRS). A

complete description of Raman amplification would approach thoroughly nonlinear

optics and complex mathematics. We choose an intermediate approach that explains

the physics of the amplifiers but keeping in mind an imediate application using a

telecommunication perspective. Thus, the work described in four major references,

Headley and Agrawal [1], J. Bromage [2]and Auyeung and Yariv [3], is followed in

order to provide a description as as relevant as possible.

The theory is approached for the first time and a brief description of spontaneous

and stimulated Raman scattering is provided in a quantum mechanical point of vue.

The Raman gain coefficient is introduced analytically and the efficiency of Raman

amplification is discussed for several types of fibers. Then, single pump amplification

is approached and the pumping architecture is discussed. The physical model to

predict the power evolution for data and pump signals is also presented. The model

is mathematically described by a set of coupled nonlinear differential equations for

which no analytical solution is available. Nevertheless, the topic of Raman amplifiers

modeling will be discussed with more detail in chapter 3. However, to provide some

physical insight, simplifications can be made, in particular, the one pump and one

signal situation small-signal approximation. An analytical approach that follows the

work developed by Auyeung and Yariv that considers pump depletion is presented,

although this method is only valid for forward single pumping.

We proceed with a description of the performance limiting factors of Raman

amplifiers, which are amplified spontaneous emission, single and double Rayleigh

backscattering, also known as multipath interference (MPI) and pump noise transfer.

Since amplified spontaneous emission is the dominant effect, it is examined with more

detail and the related concept of noise figure and effective noise figure are approched,

being their temperature dependence analysed for broadband amplifiers.

2. Raman amplification theory

2.2 Theory

In a very simplified interpretation, Raman scattering can be understand as an

inelastic scattering where light inicident on a medium is converted to a lower

frequency. The process can be described by an excitation of the molecule of the

medium up to a virtual level induced by the incident photon, followed by a de-

excitation and emission of a lower frequency photon [4]. The difference in energy

between incident and emitted photons is dissipated by molecular vibrations of the

medium. In a solid-state quantum mechanical description, optical photons are

inelastically scattered by quantized molecular vibrations called optical phonons [5],

[2]. Photon energy is lost or gained, shifting the frequency of the light. The

components of scattered light that are shifted to lower frequencies are called Stokes

lines; those shifted to higher frequencies, anti-Stokes lines. This process differs from

Rayleigh scattering, an elastic process in which the photon energy is conserved. These

processes are depicted schematically in Figure 2.1. Since, the anti-Stokes process is

typically orders of magnitude weaker than the Stokes process, it is irrelevant in the

context of optical communications and will not be mentioned again in this thesis.

Thus, from now on, when Raman scaterring is reffered, only the Stokes process is

considered.

Stokes photon

Anti-Stokes photon

Incident photon

Incident photon

photonIncident photon

Figure 2.1: Sheme describing inelastic spontaneous Raman Scattering (top) with the Sokesradiation, (middle) anti-skokes radiation. (bottom) Elastic Rayleigh radiation.

Raman scattering can also be stimulated by signal light at an appropriate frequency

shift from a pump, leading to stimulated Raman scattering. In this process, pump

and signal light are coherently coupled. In a quantum mechanical interpretation, a

pump photon is converted into a second signal photon that is an exact replica of the

first, and the remaining energy produces an optical phonon. The initial signal photon

20

2.2. Theory

is therefore amplified, as schematized in Figure 2.2. For high enough pump power,

the scattered light grows rapidly. This process is the gain mechanism behind Raman

amplification. Some aspects of it are particularly relevant, namely, it can occur in any

fiber, at any wavelength and in a very small timescale (tens of femtoseconds).

Signal photon

Pump photon

Figure 2.2: Sheme describing stimulated Raman scattering.

The most important parameter characterizing a Raman amplifiers is its gain

coefficient, γR, since it described how the signal power grows through SRS. It is related

to the imaginary part of the nonlinear third-order susceptibility, χ(3) as [6], [7]:

γR (∆ω) =4πω0

cnIm[

χ(3) (∆ω)]

(2.1)

where ω0 is the optical signal frequency, ∆ω is the frequency shift between signals and

n is the material refractive index.

An important quantity named Raman gain efficiency of the fiber, gR, relates Raman

gain coefficient with the effective area of the fiber, Aeff :

gR =γRAeff

(2.2)

The efective area can be approximated to the core area as Aeff ≈ Acore when the

field mode distribution is gaussian.

Due to the amorphous nature of Silica that allows a continuum of molecular

vibrational frequencies, the Raman gain curve is fairly broad. Larger gain coefficient

are obtained when the polarization states of the scattered light and the incident light

are parallel, being smaller values obtained for perpendicula polarizations [2]. A peak

is always observable for a frequency shift, ∆Ω, equal to 13.2 THz. When Silica is

pumped by a 1450 nm wavelength pump, the Raman gain peak is at about a 100 nm

shift ( 1550 nm). So a simple Raman amplifier can be built by choosing the signal

at Stokes shift ( 1550 nm) and pumping the Silica fiber with a 1450 nm wavelength

pump.In Figure 2.3, two Raman gain efficiency spectra are displayed, showing the

different strengths of the Raman coupling of a SSMF and a dispersion compensating

fiber (DCF), for a pump wavelength at 1450 nm. As a matter of fact, the small effective

area of the DCF (15 µm2) is determinant for its higher Raman gain efficiency when

compared to the SSMF (80 µm2) or when compared with DSF fibers (50 µm2).

21


0 5 10 15 20 25 30 35

0.0

5.0x10-4

1.0x10-3

1.5x10-3

2.0x10-3

2.5x10-3

3.0x10-3

3.5x10-3

R

aman

gai

n ef

fcie

ncy

(W-1m

-1)

pump-signal frequency difference (THz)

SSMF DCF

Figure 2.3: Raman gain coefficient spectra for two germanosilicate fibers: Standard SingleMode Fiber (SSMF) and Dispersion Compensating Fiber (DCF), for a pump wavelength of1450 nm.

2.3 Single pump amplification

In the simplest situation, the amplification system is composed by a continuous

wave (CW) pump that when launched into an optical fibre amplifies a CW signal. A

basic scheme for a RFA architecture is displayed in Figure 2.4. The signal and pump

waves are launched into the optical fiber (the gain medium) by a coupler, so, that

stimulated Raman scattering can occur. Since this effect occurs uniformly for all the

orientations between pumps and signals, Raman amplifiers can work both in forward

and/or backward pumping configuration.

The model for power evolution in Raman amplifiers with a single pump and single

signal assumes pump-to-signal and attenuation as the major interactions, stated in the

set of equations 2.3 and 2.4.

±dPs

dz= −αsPs + gRPpPs (2.3)

±dPp

dz= −αsPp −

νpνsgRPpPs (2.4)

where αp,s are the fiber attenuations measured at the pump and signal frequencies νp,s.

The term ± refers to co and counter propagating signals.

22

2.3. Single pump amplification

Forward Pump Backward pump

Fiber

Coupler Coupler

Data signals

Figure 2.4: General scheme for a distributed Raman amplifier. For simplicity the opticalisolators used to protect the pumps and signals sources, were omitted.

It is not easy to obtain an analytical solution for the set of equations 2.3 and

2.4 because it is nonlinear. However, some solutions obtained under specific

simplifications are assessable which, although not general, can provide important

information about the system behaviour.

In small signal appoximation, it is assumed that the pump depletion due to the

transfer of power to the signal (second term in equation 2.4) is negligible. This

assumption is valid whenever the signal power value is small in comparison to the

pump power. In such situation, the second equation is no longer nonlinear and thus

solvable analytically. The behaviour of the pump is an exponential decay due to

its attenuation in the fiber, as stated in equations 2.5 and 2.6 in the co and counter

propagating pumping respectively.

Pp(z) = P0 exp(−αpz) (2.5)

Pp(z) = P0 exp[−αp(L− z)] (2.6)

where P0 is the input pump power and L is the fibre length.

The substitution of the equations above in the signal differential equation allows

the derivation of an analytical solution for the signal, as follows.

Ps(z) = Ps(0) exp

(

gR

∫ z

0

Pp(z′)dz′ − αsz

)

(2.7)

Thus, the solutions at z = L are given by:

Ps(L) = Ps(0) exp (gRP0Leff − αsL) ≡ G(L) Ps(0) (2.8)

where G(L) is the net gain computed at the end of the fibre. The quantity Leff is

designated by effective length and represents the length within which most of the

amplification occurs.

23


Leff =[1− exp (−αpL)]

αp

(2.9)

Another important quantity often used is the on-off gain, Gon−off . It is defined as the

signal power increase when the pump is turned on. Therefore, in the small signal

limit, it is expressed by 2.10.

Gon−off =Psignal (z = L)with pumps on

Psignal (z = L)with pumps off≡ exp (gRP0Leff ) (2.10)

It must be noted that in both co and counter pumping the signal reaches the end

of the fibre with the same power, which means that, in terms of gain, the pumping

schemes are equivalent.

To verify the validity of the small signal approximation, a practical situation with a

pump and a travelling signal was implemented numerically. The pump is centered

at 1450 nm and has an initial power of 500 mW while the signal is centered at

1530 nm. The numerical experiment follows a copropagating amplification scheme,

being the gain medium a span of 40 km of SSMF. The implementation was carried

out with different signal input powers (1, 10 and 50 mW) using the 4th order Runge-

Kutta method with adaptative stepsize and the small signal approximation given by

equation 2.5. The implementations use values of 0.25 dB/km and 0.20 dB/km for

the pump and signals attenuation. The Raman gain cofficients were obtained for

germanosilicate fibers, using file data from the VPI software and an effective area of

80 µm2 was considered. The calculated Raman gain efficiency was also multiplied by

a factor of 1/2 because the polarizations of pump and signal are completely scrambled

along the propagation. The results are plotted on Figure 2.5.

On looking at the results, it is noticeable that the approximation stands valid when

the signal is small (1 mW) and that the effect of pump depletion becomes more evident

as the signal input power increases. The implementation of the signal power evolution

also demonstrates the limitations of the small signal approximation, as shown on

Figure 2.6, where the small signal approximation obtained with equation 2.7 was

plotted together with the numerical one, for an input signal power equal to 10 mW.

Indeed, in the numerical solution, the signal reaches the end of the optical fiber with

more power due to the depletion exerted on the pump. However, the small signal

approximation is useful when its applicability was previously verified.

In the bidirectional pumping, some modifications to the system of equations 2.3

and 2.4 have to be made, due to the addition of an extra pump. In this situation,

the total inputted pump power is still P0 but a fraction of it goes in each direction.

Considering the small signal approximation, the equations are given by:

dP+p

dz= −αpP

+p (2.11)

24


0 10000 20000 30000 40000

0.0

0.1

0.2

0.3

0.4

0.5

Pum

p po

wer

(W)

Distance (m)

small signal approximation numerical, input signal=1 mW numerical, input signal=10 mW numerical, input signal=50 mW

Figure 2.5: Pump power evolution along 40 km of SSMF in the small signal approximation andconsidering pump depletion for an input signal power equal to 1 mW, 10 mW and 50 mW.

0 10000 20000 30000 400000.008

0.010

0.012

0.014

0.016

0.018

0.020

0.022

0.024

0.026

0.028

Sig

nal p

ower

(mW

)

Distance (m)

numerical small signal approximation

Figure 2.6: Signal power evolution along 40 km of SSMF in the small signal approximationand considering pump depletion. The initial pump power is equal to 500 mW.

dP−p

dz= αpP

−p (2.12)

25


Pp = P+p + P−

p (2.13)

with P+0 = r+P0 and P−

0 = (1− r+)P0.

The solution of the system of equations 2.11, 2.12 and 2.13 for the total pumping

0 10000 20000 30000 40000

0.16

0.18

0.20

0.22

0.24

0.26

0.28

Tota

l pum

p po

wer

(mW

)

Distance (m)

small signal approximation numerical, input signal =1 mW numerical, input signal = 10 mW numerical, input signal = 50 mW

Figure 2.7: Pump power evolution along 40 km of SSMF in the small signal approximation andconsidering pump depletion for an input signal power equal to 1 mW, 10 mW and 50 mW.

power is given by a combination of the solutions in the co and counter-propagating

directions, as follows:

Pp(z) = P0

[

r+exp (−αpz) +(

1− r+)

exp (−αp (L− z))]

(2.14)

The substitution of equation 2.14 in equation of 2.3 yields to the solution of the

signal for bidirectional pumping.

To verify the validity of the small signal approximation in the bidirectional scheme,

the situation described above was implemented but two bidirectional pumps centered

at 1450 nm were used instead of one. The total inputted pump power was still equal

to 500 mW (250 mW in each end of the fiber) and a signal centered at 1550 nm was

used. Once again, the initial signal powers were tested for 1 mW, 10 mW and 50 mW.

The numerical implementation follows the APA method which will be presented more

detailly in next chapter. The numerical results were compared with the ones obtained

recurring to equations 2.11, 2.12 and 2.13 and plotted in Figure 2.7. On looking

at the results, it is noticeable that unlike the co-propagated pumping situation the

approximation stands valid for the three analyzed cases.

26


2.3.1 Pump depletion

The analytical results provided by the small signal approximation are based on

the idea that the pump depletion is negligible, otherwise, the system of equations 2.3

and 2.4 is commonly solvable by numeric methods. However, in the co-propagated

pumping architecture, the model developed by Auyeung et al [3] provides a suitable

analytical solution and thus will be discussed in frame of this thesis.

As said above, Auyeung model for stimulated Raman scattering is valid for co-

propagating single pump and multi-signal situation, however, only the single signal

is analyzed here. It accounts for pump depletion and spontaneous scattering being its

results broad enough to contemplate the small signal approximation presented above.

The system of equations 2.3 and 2.4 may be rewritten in terms of the pump and

signal photon numbers np and ns as [8]:

dns

dz= −αsns + γ0npns + γ0ns (2.15)

dnp

dz= −αpnp − γ0npns − γ0np (2.16)

where γ0 is the nonlinear Raman gain constant.

It must be noted that the last term in the pump and signal equations is related

to the spontaneous scattering. The total number of photons, n0, is conserved in this

process, and may be computed recurring to the condition at z = 0, as:

n0 = np0 + ns0 (2.17)

where np0 and ns0 are the number of pump and signal photons at z = 0.

Making the approximation that pump and signal suffer the same attenuation α

(this assumption is especially true in the infrared region where the attenuation is

near its minimum and thus its variation with wavelength is small), the addition of

equations 2.16 and 2.15 is given by:

d

dz(np + ns) + α(np + ns) = 0 (2.18)

Hence, the total number of photons varies as:

np + ns = n0 exp (−αz) (2.19)

By replacing 2.19 in 2.15 and 2.16, the following is obtained:

dnp

dz+ np [α + γ0 + γ0n0 exp (−αz)] = γ0n

2p (2.20)

27


dns

dz+ ns [αγ0 − γ0n0 exp (−αz)] = −γ0n

2s + γ0n0 exp (−αz) (2.21)

It must be noted that the system of equations 2.20 and 2.21 together with the initial

conditions 2.17 is now solvable analytically because it is composed by first order exact

differential equations. Thus, taking a change in the variable np = 1/x, the equation for

the pump number of photons is converted into:

dx

dz− x [α + γ0 + n0 exp (−αz)] = −γ0 (2.22)

Using the integrating factor exp [− (α + γ0) + γ0n0/α exp (−αz)])], the equation takes

the form:

x =exp[

(α + γ0) z −γ0n0

αexp (−αz)

]

+

+

∫ z

0

[

−γ0 exp ((α + γ0) z′) +

γ0n0

αexp (−αz′)

]

dz′ + C (2.23)

where C is a constant which will be determined by the initial conditions. Solving

the integral with the integration by parts technique, the solution is given by:

x =1

n0

exp (αz)

1− γ0 exp[

γ0z −γ0n0

αexp (−αz)

]

∞∑

l=0

(

γ0n0

α

)lexp [− (lα + γ0)]

(lα− γ0)!

+

+ C exp[

(α + γ0z)−γ0n0

αexp (−αz)

]

(2.24)

Finally, the pump photon number is given by the following:

np =n0 exp (−αz)

1 + η + ξ(2.25)

where

η = γ0 exp[

γ0z −γ0n0

αexp (−αz)

]

∞∑

l=0

(

γ0n0

α

)lexp [− (lα + γ0)]

(lα− γ0)!(2.26)

ξ =ns0

np0

exp

γ0z +γ0n0

α[1− exp (−αz)]

(2.27)

The determination of the photon signal equation is trivial taking into account equation

2.19 and can be expressed by:

ns = n0 exp (−αz)η + ξ

1 + η + ξ(2.28)

28

2.4. Performance limiting factors

The solutions for the pump and signal photon numbers can be converted to pump and

signal power according to:

Pp (z) =P0 exp (−αz)

1 + η + ξ(2.29)

Ps (z) =νsνpP0 exp (−αz)

η + ξ

1 + η + ξ(2.30)

η =νpνs

hνsvgP0L

exp

gRP0

αAeff

[1− exp (−αz]

)

(2.31)

ξ =νpνs

Ps0

Pp0

exp

gRP0

αAeff

[1− exp (−αz]

)

(2.32)

The limiting situations are assessed taking the limits ξ + ν << 1 and ξ + ν >> 1.

For the first situation, the pump power evolution is compliant with the small signal

approximation, being the fiber attenuation the dominant effect. For the second

situation, a full conversion of photons from pump to signal is achieved and thus the

signal power evolution is given by:

Ps (z) ≈νsνpP0 exp (−αz) (2.33)

2.4 Performance limiting factors

Although the benefits of Raman amplification are valuable and important in fiber

optics communication systems, there are concerns to be aware of. The limitations

on the performance of Raman amplifiers arise from three fundamental contributions:

spontaneous Raman scattering, double Rayleigh backscattering and pump noise

transfer. Other effects derived from polarization mode dispersion [9], [10] are also

detrimental but will not be approached in the scope of this thesis.

Spontaneous Raman scattering and its subsequent amplification is dominant effect

of noise in Raman amplification.

2.4.1 Amplified spontaneous emission (ASE)

The noise added from the spontaneous Raman scattering is ougthed to the random

phases of the photons released in the process. The generation and amplification of ASE

obeys:

±dP±

ASE

dz= −αASEP

±ASE + gRPpP

±ASE + gR [1 + η (T )]hνASEBrefPp (2.34)

29


where P±ASE is the ASE power in a bandwidth Bref , propagating in the direction ±.

One difference between this equation and equation 2.3 for the signal is the

additional inhomogeneous source term in the differential equation. This source term

includes a phonon occupancy factor, η (T ):

ηij (T ) =

[

exp

(

h∆ν)

kBT− 1

)]−1

(2.35)

where h is the Planck constant, kB is Boltzmann constant, T is the absolute temperature

of the fiber in kelvins, and ∆ν is the frequency separation between the pump and

signal. The boundary conditions set that P+ASE (z = 0) = 0 and P−

ASE (z = L) = 0,

however, initial values can be assumed when the pump is not purely monocromatic

[11].

It has been demonstrated that the amount of accumulated ASE depends on

the amplification scheme. Typically copropagated pumping schemes are less noisy

because most of the Raman gain is concentrated in the fiber input; however the

counterpropagated pumping is often employed in practice because of the transfer of

the pump noise transfer This issues is discussed later.

The noise figure of an optical amplifier amounts the degradation of the optical

signal to noise ratio when the signals are amplified. As said above, the most important

source of noise in optical amplifiers is ASE, which, for Raman amplifiers is due to

spontaneous scattering. Assuming that the signals are initially as noiseless as possible,

and that their degradation is due to signals spontaneous beat noise produced by

ASE [12], the noise figure, in linear units, is given by [2], [13], :

NF ≈

(

2P+ASE (z = L)

hνASEBref

+ 1

)

1

G(2.36)

where G is the net gain. The first term corresponds to the noise from the signal

spontaneous beating and the second one to shot noise.

Another quantity, named effective noise figure, accounts the noise figure that a

discrete amplifier placed at the end of an unpumped fiber link would need to provide

the same noise performance that a distributed Raman amplifier. In decibel units, the

effective noise figure is computed using:

NF dBeff = NF dB − αLdB (2.37)

The spectra depicted in Figure 2.8 represent the net gain of effective noise figure of

a distributed RFA composed by two bidirectional pumps at 1440 nm and 1500 nm with

a total power of 1 W (500 mW each) and and 25×400 GHz data signals along a span of

50 km SSMF. The results were obtained by simulation using the collocation method.

30


The simulation had considered attenuation coefficients of 0.25 dB/km and 0.22 dB/km

for pumps and data signals respectively. The step size is adaptative. On looking at

1500 1510 1520 1530 1540 1550 1560 1570 1580 1590-4

-3

-2

-1

0

1

2

3

4

wavelength (nm)

Net

gai

n (d

B)

-1

0

1

2

3

4

5

Effective noise figure (dB

)

Figure 2.8: Net gain and effective noise figure spectra for a system with two bidirectionalpumps and 25× 400 GHz data signals along a span of 50 km SSMF.

the noise figure spectrum, it is noticeable that there is a decreasing tilt with frequency.

This effect is due to the different amount of gain experiencied by the data signals

from the pump. The longest wavelength pump signal provides more gain the longest

wavelength data signals and is also amplified by the shorter wavelength pumps. As

a result, the is gain is more evenly distributed along the fiber and the noise figure has

a better performance. Additionally, as the frequency separation between pump and

shorter wavelength data signals decreases, the amount of generated ASE is higher.

This tilt is also dependent on temperature, as explained below.

The performance of Raman amplifiers in terms of low noise is determined by

bandwidth, as demonstrated by Fludger et al [14], where a fundamental compromise

must be founded. Thus, The quantum limit noise figure is not obtained at finite

temperature at room temperature, showing variations at shorter wavelengths due to

the increase in spontaneous wavelengths.

Using the undepleted pump approximation and considering that pump and data

signals have the same attenuation, the net gain in linear unit is given by:

G = exp

(

−αL+ Leff

∑

i

gRiPpi

)

= exp (−αL)∏

i

gi (2.38)

31


where L is the fiber length and Leff its effective length, α is the attenuation, gRi is the

Raman gain efficiency for a data signal interacting the ith pump signal. The amplifier

spontaneous emission (ASE) at the fiber output is given by [15]:

PASE (L) = hν

[∑

i Ei ln (gi)∑

i ln (gi)

](

1 +αLeff exp (αL)∑

i ln (gi)

)

G−

(

1 +αLeff∑

i ln (gi)

)

(2.39)

where h is the Planck constant and Ei is is a Bose distribution that takes into account

the thermal population of phonons in the ground state [16], [17], given by:

Ei = 1 + η (T ) (2.40)

The noise figure given by equation 2.36 can be approximated by the following

expression:

NF ≈ 2

∑

i Ei ln (gi)∑

i ln (gi)(2.41)

Equations 2.38 and 2.41 were implemented for a broadband Raman amplifier with

a spectral width of 120 nm. The data signals are amplified by means of five backward

pump signals at 1402 nm, 1423 nm, 1443 nm, 1464 nm and 1495 nm, each of them

with an input power of 125 mW in a span of 25 km of SSMF. The Raman gain

efficiency values were obtained from Figure 2.3 and an attenuation of 0.22 dB/km was

considered for pumps and data signals. The results of the noise figure for at absolute

temperatures of 300 K, 200 K, 100 K and 0 K are displayed in Figure 2.9.

The results of the noise figure spectra exhibit a decaying on the ripple as the

temperature approaches the 0 K. The noise figure of shorter wavelengths data signals

is higher than the one observed at larger wavelengths because the spontaneous

emission increases as the signal wavelengths approach the pump wavelengths. At

lower temperatures, the phonon distribution is reduced and the occupancy factor

tends to one, reducing the noise figure to 3 dB, the value defined by the theoretical

quantum limit [18].

32


1500 1520 1540 1560 1580 1600

3.0

3.2

3.4

3.6

3.8

4.0

4.2

4.4

4.6

Noi

se F

igur

e (d

B)

Wavelength (nm)

300 K 200 K 100 K 0 K

Figure 2.9: Noise Figure spectra for temperatures equal to 300 K, 200 K, 100 K and 0 K.

2.4.2 Multipath interference

The double Rayleigh backscattering, also known as MPI is a phenomenon that

limits the performance of distributed Raman amplifiers. It consists of an elastic

reflection of light in the sub-micron inhomogeneities in the fiber refractive index.

When the signal is backscattered twice and experience significant gain it can create

crosstalk in the copropagating direction [19], appearing as a noise source. Thus this

fenomenon sets a fundamental limit on the maximum amount of distributed Raman

gain that can be used before MPI dominates. The simulation of this effects can be

implemented using the following equations [20]:

−dPSRB

dz= −αsPSRB + gRPpPSRB + frαrPs (2.42)

dPDRB

dz= −αsPDRB + gRPpPDRB + frαrPSRB (2.43)

where PSRB and PDRB represent average power levels of noise created by single and

double Rayleigh backscattering, respectively, αr is Rayleigh scattering loss and fr the

fraction that is recaptured by the fiber mode.

In the small signal approximation approach, the fraction of the input power

coming up with the data signal, fDRB = PDRB/Ps, is given by the following equation

33


[21]:

fDRB = (frαr)2

∫ L

0

G−2 (z) dz

∫ L

z

G2 (z′) dz′ (2.44)

The crosstalk originated by multipath interference can lead to large power

penalties, especially in long haul systems where this crosstalk accumulates over

multiple amplifiers.

2.4.3 Pump noise transfer

Pump noise transfer is fundamentally due to fluctuations in the pump power. Since

the Raman effect acts very fast (in a femtosecond time scale) those fluctuations of

power are transferred to the signal as noise. In distributed Raman amplifiers, the gain

is built as the waves travel in the fiber. This produces an averaging effect that limits

the bandwidth over which pump noise is transferred to the signals. This phenomenon

can be described in the frequency domain as a transfer of relative intensity noise

(RIN) from a pump to a signal, using a frequency-dependent transfer function, H (f),

that depends on the pumping direction, the pump and signal wavelengths, and the

amount of on-off Raman gain [22].

RINs (f) = H (f)RINp (f) (2.45)

In a forward pumping scheme with low dispersion, signal and pump are traveling

together and any power fluctuation of the pump is perceived by the signal in the same

temporal window. However, for the case of backward pump the signal moves out of

the temporal window of the pump and perceived an average gain. The same effects

could be obtained in a forward pumping scheme if the fiber dispersion is sufficiently

high to separate temporally the pump and signal waves.

The amount of Q-factor degradation for a given amplifier design can be estimated

using [22]:

∆QdB = 10 log10

(

1 +Qs

∫ f2

f1

RINs (f) df

)

(2.46)

where Qsis the reference Q-factor in linear units without RIN transfer, while f1 and f2lower and upper frequency limits of the receiver.

2.5 Chapter summary

This chapter described the theory that suppors Raman fiber amplifiers. As

said above, the thoroughness of the approached theory is the stricly necessary for

the understanding of the work that was carried one in the subsequent chapters.

Thus, spontaneous and stimulated Raman scattering are explained as inelastic

34

2.5. References

scatterings with the generation of Stokes and anti-Stokes radiation. We proceeded

with the analysis of simpler cases, namely the small signal approximation where

pump depletion is neglected in order to obtain analytical solutions. An analytical

approach of pump depletion is also presented for a simple case of single forward

pumping. Detrimental effects that impair the performance of Raman amplifiers are

then presented and analyzed for broadband Raman amplifiers in order to understand

the architectures that are currently used in modern communication systems.

References





[3] J. AuYeung and A. Yariv, “Spontaneous and stimulated Raman scattering in long

low loss fibers,” IEEE Journal of Quantum Electronics, vol. 14, no. 5, pp. 347–352,

1978.

[4] G. Agrawal, Nonlinear Fiber Optics. Elsevier Academic Press, 2007.

[5] C. Kittel et al., Introduction to solid state physics. Wiley New York, 1996.

[6] A. Atieh, P. Myslinski, J. Chrostowski, and P. Galko, “Measuring the Raman Time

Constant (T_R) for Soliton Pulses in Standard Single-Mode Fiber,” J. Lightwave

Technol, vol. 17, pp. 17–2, 1999.

[7] P. Andre, R. Correia, L. Borghesi, A. Teixeira, R. Nogueira, M. Lima,

H. Kalinowski, F. Da Rocha, and J. Pinto, “Raman gain characterization in

standard single mode optical fibres for optical simulation purposes,” Optica

Applicata, vol. 33, no. 4, pp. 559–574, 2003.

[8] A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE Journal of

Quantum Electronics, vol. 9, no. 9, pp. 919–933, 1973.

[9] D. Wang and C. Menyuk, “Calculation of Penalties Due to Polarization Effects in

a Long-Haul WDM System Using a StokesParameter Model,” Journal of Lightwave

Technology, vol. 19, no. 4, p. 487, 2001.

[10] G. Foschini and C. Poole, “Statistical theory of polarization dispersion in single

mode fibers,” Lightwave Technology, Journal of, vol. 9, no. 11, pp. 1439–1456, 1991.

35


[11] A. Rocha, B. Neto, M. Facao, and P. Andre, “Study of raman amplification with


no. 2, pp. 301–303, 2008.

[12] N. A. Olsson, “Lightwave systems with optical amplifiers,” Lightwave Technology,

Journal of, vol. 7, no. 7, pp. 1071–1082, 1989.

[13] D. Derickson, Fiber Optic Test and Measurement. Packard professional books,

Prentice Hall PTR, Upper Saddle River, New Jersey, 1998.

[14] C. Fludger, V. Handerek, N. Jolley, and R. Mears, “Fundamental noise

limits in broadband raman amplifiers,” in Optical Fiber Communication

Conference. Optical Society of America, 2001, p. MA5. [Online]. Available:

http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2001-MA5

[15] S. R. Chinn, “Analysis of counter-pumped small-signal fibre Raman amplifiers,”

Electronics Letters, vol. 33, no. 7, pp. 607–608, 1997.

[16] M. A. Farahani and T. Gogolla, “Spontaneous raman scattering in optical

fibers with modulated probe light for distributed temperature raman remote

sensing,” J. Lightwave Technol., vol. 17, no. 8, p. 1379, 1999. [Online]. Available:

http://jlt.osa.org/abstract.cfm?URI=JLT-17-8-1379

[17] K. Rottwitt, M. Nissov, and F. Kerfoot, “Detailed analysis of Raman amplifiers

for long haul transmission,” in Optical Fiber Communication Conference and Exhibit,

1998. OFC’98., Technical Digest, 1998, pp. 30–31.


York, 1994.

[19] C. Kim, J. Bromage, and R. Jopson, “Reflection-induced penalty in Raman

amplified systems,” IEEE Photonics Technology Letters, vol. 14, no. 4, pp. 573–575,

2002.

[20] J. Bromage, P. Winzer, and R. Essiambre, “Multiple path interference and its

impact on system design,” Raman Amplifiers for Telecommunications 2, pp. 491–568,

2004.

[21] M. Nissov, K. Rottwitt, H. Kidorf, M. Ma, T. Ltd, and N. Eatintown, “Rayleigh

crosstalk in long cascades of distributed unsaturated Raman amplifiers,”


[22] C. R. S. Fludger, V. Handerek, and R. J. Mears, “Pump to signal RIN transfer in

Raman fiber amplifiers,” Lightwave Technology, Journal of, vol. 19, pp. 1140–1148,

2001.

36

http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2001-MA5

http://jlt.osa.org/abstract.cfm?URI=JLT-17-8-1379

Chapter 3

Modeling Raman fiber amplifiers

3.1 Introduction

The most accurate and straightforward approach in designing a RFA is solving the

Raman amplifier propagation equations. The practical application of this approach,

however, is limited by its low efficiency because the set of ODEs is nonlinear. The

lack of efficiency is also affected by the fact that RFAs being traditionally operated in

the backward-pumping geometry to minimize the polarization dependent gain and

the noise induced by the fluctuation of the pumps power [1], [2]. With this geometry,

the coupled differential equations of RFAs become boundary value problems (BVPs),

which are much more difficult to be solved than the initial value problem (IVP)

involved in the forward-pumping situation.

The methodologies used to solve the set of Raman propagation equations

rely essentially on numerical methods. Depending on the chosen method, the

solutions are accurate although not necessarily efficient because their computational

implementation is usually time consumming. However, some qualitative insigths

about the solutions behaviour can be provided by means of stability analysis.

Semi-analytical methods can improve significantly the efficiency of the solutions

computation, such as APA.

This chapter is concerned with the simulation of Raman fiber Amplifiers and

several methods will be discussed. These methods can be either numerical (shooting,

collocation) or semi-analytical (APA) but no analytical solution is available due to the

nonlinearity of the ordinary differential equations that describes the system. However,

qualitative analysis can provide deeper insigths about the solutions behaviour in the

vicinity of critical points. Thus, an analysis for Raman propagation equations near

their critical points was carried out by linearization and also using Lyapunov second

method. The only critical point with physical significance was identified as the origin

of the phase plane and demonstrated to be asymptotically stable. Additionally some

trajectories were also plotted in the phase plane.

3. Modeling Raman fiber amplifiers

3.2 Mathematical model

The model for power evolution in Raman amplifiers assuming, a multipump

multisignal configuration is often based on an unified treatment of channels, pumps

and spectral components of ASE. The major interactions can be reasonably drawn

by considering the pump-to-pump, signal-to-signal and pump-to-signal power

transfer, attenuation, Rayleigh backscattering, spontaneous Raman emission and its

temperature dependence as well as the ASE. Other effects, such as noise generation

due to spontaneous anti-Stokes scattering, polarization and nonlinear index are

neglected. It must be noted that signal channels and Raman pumps are treated as

fields at single frequencies, so ignoring the interactions due to the spectral shape of

signals and pumps.

In a general approach, the power evolution of pumps, signals and ASE (in forward

and backward directions), with time along the fiber distance is given by the following

set of coupled differential equation [3]. For Np pumps, Ns probe signals and NASE

spectral components for ASE, the system is formed by Np +Ns + 2NASE equations.

∂P±i (z, t)

∂z∓

1

Vi

∂P±i (z, t)

∂t=∓ αiP

±i (z, t)± γiP

±i (z, t)

± P±i (z, t)

i−1∑

j=1

gji[

P+j (z, t) + P−

j (z, t)]

∓ P±i (z, t)

m∑

j=i+1

νiνjgij[

P+j (z, t) + P−

j (z, t)]

∓ 2hνi

i−1∑

j=1

gji[

P+j (z, t) + P−

j (z, t)]

(1 + ηji)∆ν

∓ 4hνiP±i (z, t)

m∑

j=i+1

gij (1 + ηij)∆ν (3.1)

Vi is the frequency dependent group velocity. The ± signs stand for the forward

or backward propagating waves, respectively, being αi and γi the attenuation and

Rayleigh coefficients of the ith wave at frequency νi. The phonon occupancy factor is

given by:

ηij =

[

exp

(

h (νi − νj)

kBT− 1

)]−1

(3.2)

where h and kB are the Planck and Boltzmann constants, respectively, and T is the

fiber absolute temperature.

The frequencies νi are numbered by their decreasing value (the lower order

corresponds to the higher frequency). By this way, the terms in the summation in

38

3.2. Mathematical model

equation 3.1, from j = 1 to j = i − 1 cause amplification since the wave i is receiving

power from the lower order waves (with higher frequency). For the same reason,

the terms in the summation from j = i + 1 to j = m originate attenuation. For

mathematical convenience the gain spectrum was divided into spectral slices of width

∆ν, spanning the range over which ASE spectral components are significant.

The terms that contain a product of powers describe the coupling via stimulated

Raman Scattering, being its strength determined by the Raman gain efficiency of the

fiber, gij obtained by:

gij =γR (νi − νj)

ΓAeff

(3.3)

where Aeff is the effective area of the fiber and the factor Γ is a dimensionless

quantity comprised between 1 and 2 that takes into account the polarization random

effects. The achieved gain, as well as the slope of the gain spectrum, depends on the

transmission fiber [4], [5] as depicted in Figure 2.3.

To obtain a steady-state power distribution, the time derivative in 3.1 is settled

equal to zero, and the set of equation takes the form of.

dP±i (z)

dz=∓ αiP

±i (z)± γiP

±i (z)

± P±i (z)

i−1∑

j=1

gji[

P+j (z) + P−

j (z)]

∓ P±i (z)

m∑

j=i+1

νiνjgij[

P+j (z) + P−

j (z)]

∓ 2hνi

i−1∑

j=1

gji[

P+j (z) + P−

j (z)]

(1 + ηji)∆ν

∓ 4hνiP±i (z)

m∑

j=i+1

gij (1 + ηij)∆ν (3.4)

If the purpose of the simulation is to obtain only a gain profile [6], spontaneous

Raman scattering and Rayleigh backscattering can be ignored in 3.4, and the system

of equations become:

39


dP±i (z)

dz=∓ αiP

±i (z)

± P±i (z)

i−1∑

j=1

gji[

P+j (z) + P−

j (z)]

∓ P±i (z)

m∑

j=i+1

νiνjgij[

P+j (z) + P−

j (z)]

(3.5)

3.3 Numerical approaches

Basically, there are two distinct classes of numerical methods for solving BVP:

shooting methods and relaxation methods [7]. The set of ODEs, which cannot be

solved analytically have also to be implemented numerically through well known

methods such as Runge-Kutta (RK), Adams-Bashforth (AB), Adams-Bashforth-

Moulton (ABM), among others. However, for the specific application of Raman

propagation equations, novel approaches have been devoloped, in order to provide

fast and reliable solutions [8]. These techniques which are based on the above

referred methods are able to support BVP solvers, such as shooting, by increasing

their accuracy and efficiency. Since a complete revision of those numerical methods is

beyond the scope of this thesis, only some example are provided, namely shooting

method with enahnced Runge-Kutta developed by Xueming Liu [9]. Additonally,

collocation method is presented and its application to Raman amplifiers through the

Matlab solver bvp4c described extensively.

3.3.1 Shooting Method

In direct shooting methods, we choose values for all the dependent variables at

one boundary, solve the system of ODEs as an IVP and verify if the obtained values

on the other boundary are consistent with the stipulated values (boundary conditions)

[7], [10], [11]. The IVP integration can be carried out by the fourth order Runge-Kutta

methods with adaptative step-size, taking into account that its absolute stability region

is (-2.785, 0) where Wilf criterion is fulfilled [12]. Then, the parameters are repeatedly

changed using some correction scheme until this goal is attained. The selection of the

correction scheme is crucial for stability and efficiency of the resulting algorithm [7].

The basic difficulty to shooting is that a perfectly nice BVP can require the integration

of IVP that are unstable. The solution of a BVP can be insensitive to changes in

boundary values, yet the solution of the IVP of shooting are sensitive to changes in

40

3.3. Numerical approaches

initial values.

The work developed by Liu and Lee [9] use direct shooting to solve backward

or bidirectional pumping Raman amplifiers but the converted IVP is solved by an

enhanced fourth-order Runge-Kutta method. This method was developed to promote

faster convergence avoiding the solutions to blow up before reaching the end of the

fiber. The method is described bellow.

Considering that an initial value problem can be written as:

dx

dt= f (t, x) (3.6)

where x is a vector given by x = [x1 (t) , x2 (t) , x3 (t) , . . .] with the following initial

conditions:

x (t = 0) = x0 (3.7)

Numerical solutions based on Runge-Kutta (RK) are based on the iterative

implementation of 3.8 and 3.9:

xn+1 = xn +m∑

i=1

wiki (3.8)

ki = hf

(

tn + ci, xn +m∑

j=1

dijkj

)

(3.9)

where wi, ci, dij denote all the parameters and h the spatial step.

By introducing y = ln x and g (t, x) = f (t, x) /x, equation 3.6 becomes:

dy

dt= g (t, exp (y)) (3.10)

According to the operational process of 3.8 and 3.9, the system of ODE can be

solved as:

yn+1 = yn +m∑

i=1

wiqi (3.11)

qi = hg

(

tn + ci, exp (yn)

(

1 +m∑

j=1

dijqj

))

(3.12)

Returning to the variable x and using the fourth-order Runge-Kutta method, the

numerical implementation is given by 3.13 to 3.18:

xn+1 = xn exp [(q1 + 2q2 + 2q3 + q4)h/6] (3.13)

41


q1 = g (tn, xn) (3.14)

q2 = g (tn + h/2, xn + xnq1h/2) (3.15)

q3 = g (tn + h/2, xn + xnq2h/2) (3.16)

q4 = g (tn + h, xn + xnq3h) (3.17)

g (tn) = f (tn) /xn (3.18)

The exponential change of variable x = exp (y) is advantageous in amplifier

simulations because light amplification or attenuation resembles the exponential

pattern with propagation distance. Thus, such a change of variable may lead to faster

convergence than some linear or polynomial approximation.

The implementaion of the direct shooting using this enhanced Runge-Kutta

formula, allows the solutions to reach the fiber end without crashing.

To demonstrate those results, a Raman amplifier with one 1550 nm probe signal

with 1 mW of initial power and one 1450 nm backward pump signal with 0.5 W

of initial power was implemented using direct shooting method with both fourth

order Runge-Kutta (RK4) methods. The Raman efficiency was obtained from the

SSMF values plotted in Figure 2.3. The solution that uses simple RK4 crashed at a

maximum fiber length equal to 38 km, but the solution misleading behaviour was

already assessible, as depicted in Figure 3.1 where the pump power solutions obatined

through both methods along the fiber are plotted.

42

3.3. Numerical approaches

-5 0 5 10 15 20 25 30 35 40

0.0

0.1

0.2

0.3

0.4

0.5

Pum

p po

wer

(W)

Fiber length (km)

Enhanced RK4 Simple RK4

Figure 3.1: Pump power solutions obtained through direct shooting along the fiber length.(solid line) - Enhanced RK4, (dashed line) - Simple RK4.

Recently, some shooting algorithms with different correction schemes for the

design of Raman fiber amplifiers have been proposed in order to improve convergence

of the solutions even for larger fiber lengths [8]. This scheme is obtained by modifying

the numerical method used to perform the IVP integration (fourth order Runge-

Kutta, Runge-Kutta-Felhberg, etc [11]). In other variant of the shooting method,

we can guess boundary values at both ends of the domain, integrate the equation

to a common midpoint and repeatedly adjust the guessed boundary values so that

the solution tends to the same value at the middle point. This adjustment task is

usually performed by the Newton-Raphson method. Other approaches propose a

shooting method to a fitting point using a correction scheme based on a modified

Newton approach [13]. Therefore, by introducing the Broydens rank-one method [14]

into the modified Newton method, the algorithm becomes more efficient and stable.

This happens because the intensive numerical calculations of the Jacobi matrix are

substituted by simpler algebraic calculations [13].

3.3.2 Collocation Method

An alternative to shooting methods can rely on expansion methods, namely

collocation. Unlike shooting, collocation method approximate the solution over the

whole interval of integration on a mesh of collocation points, zi. The solution of the

differential equations are then approximated by a polynomial in each sub-intervals

[zi, zi+1] where the boundary conditions in the midpoints and extremities must be

43


satisfied. The BVP becomes a system of algebraic non-linear equations that can be

solved iteratively. It is important to remark that the resulting polynomials must be

continuous functions in each sub-intervals and therefore the residuals are forced to

become optimally zero or to reach an acceptable minimum [15], [16], [17].

The collocation method can be implemented by a MATLAB, bvp4c, which

implements the collocation method for BVP in the interval [z = 0, z = L]. The function

bvp4c is composed by three parts: the function odes to describe the ODEs, the function

bcs to calculate the residuals and the function bvpinit to implement the mesh of points.

An initial hint for the solution guess is also provided to start the iterations [18].

The needed computation time to implement bvp4c will depend mostly on the

number of points on the mesh. The solver will provide a number of 10 points by

default, however depending on the problem under study, other mesh configuration

could be more benefitial. Another important issue rely on the fact that the algorithm

can provide more than one solution compliant with the boundary conditions. So it

is important to supply the solver with an initial hypothesis that can evolve to the

desired solution. This guessed solution, given by guess, needs to take into account

the behaviour of the system. Establishing this initial solution requires a particular

knowledge of the problem and is definitely the most difficult task on using this

method. To implement the solution of Raman propagation equations, the file odes

is used to describe the sustem of ODEs, while the hint for the pumps was provided

considering the small signal approximation situation and thus it includes only the

effect of attenuation. For the transmission channels, the hint is a constant function

equal to the initial power.

P+b (zi) =

(

P+b

)

0exp (−αzi) (3.19)

P−b (zi) =

(

P−b

)

0exp (−α (L− zi)) (3.20)

Pt (zi) = (Pt)0 (3.21)

The solver bvp4c also controls the residuals in each iterations. The solution is found

when the latter are uniform in all the points on the mesh and below 1× 10−100.

3.4 Stability analysis

An alternative to traditional numerical methods can rely on analytical approaches.

Unlike numerical methods, a stability analysis can provide additional insights about

the behaviour of the solutions in terms of trajectories in phase plane and in the

vivinity of critical points. This knowledge is potentially very helful when a general

understanding of the solutions is required.

44

3.4. Stability analysis

The stability analysis of nonlinear systems is a mathematical formalism that

allows the acquisition of qualitative information about the solution rather than the

solutions itself. The methods employed are essentially geometrical making use of

the representation of the solutions trajectories in the phase plane. These trajectories

provide important information about the solutions behaviour, namely in asymptotic

conditions and in the vicinity of the critical points, the ones where the derivatives are

set equal to zero.

Consider a general autonomous system with the form dx/dt = A~x, where A is a

2× 2 constant matrix and x is a 2× 1 vector. The solution of such system has the form

x = ξert, which by direct substituion origins:

(A − rI) ξ = 0 (3.22)

Thus, r is an eigenvalue and ξ the corresponding eigenvector of the coefficient matrix

A. The eigenvalues are the roots of the following polynomial equation:

det (A − rI) = 0 (3.23)

The critical points of the system are obtained making Ax = 0 being the

corresponding solutions constant on those points. According to the eigenvalues some

assumptions about the stability of the critical point can be inferred, as summarized

in Table 3.1. If the matrix A is singular, its determinant will not vanish and the only

critical point is x = 0.

Table 3.1: Stability properties of linear systems.

Eigenvalues Type of critical point Stability

r1 > r2 > 0 Node Unstabler1 < r2 < 0 Node Asymptotically stabler1 = r2 > 0 Saddle point Unstabler1 = r2 < 0 Proper or improper node Unstable

r1, r2 = λ+ iµ Proper or improper node Asymptotically stableλ > 0 Spiral point Unstableµ < 0 Spiral point Asymptotically stable

r1 = iµ, r2 = −iµ Center Stable

A general solution of equation 3.23 can be viewed as a parametric representation

of a curve in the x1x2 plane, the so called phase plane. The shape of the curve in

the vicinity of the critical point depends on the eigenvalues of the matrix and gives

qualitative information about the systems stability [19], [20].

For the situation of nonlinear differential equations, qualitative information

around the system critical points can still be inferred by linearizing the system. The

45


center manifold theorem is a powerful tool to analyze the dynamics of nonlinear

systems near equilibrium points and states that associated with an equilibrium point

there exists an invariant manifold containing that point. The manifold is unique for

stable and unstable equilibrium points [21]. Therefore, it is possible to determine the

eigenvalues in order to classify the system stability and also obtain analytical solutions

that are valid in the vicinity of the critical point.

By looking at equations 3.5 , we notice that the origin (Pi = 0) is an equilibrium

point of system independently of the number of waves m. The stability of the

equilibrium point can be determined by the stability of the following equivalent

linearized system [22]:

±dPi

dz= JijPj (3.24)

where J is the Jacobi matrix.

Concerning the study of Raman equations, our study will be previously focused

on the simplest case, the one signal, one forward pump situation, for a matter of

simplicity. Thus, the system is given by the set of equations 2.3 and 2.4. By setting

the derivatives equal to zero, we obtain the following critical points:

(

P cs , P

cp

)

= (0, 0) (3.25)

(

P cs , P

cp

)

=

(

−αsνsgRνp

,αp

gR

)

(3.26)

Since the second critical point has a negative abscissa and both probe and pump

signals powers must be positive values, the only point physically relevant is the origin

of the phase plane.

To examine the local behavior of the solution near the critical point, we can neglect

the nonlinear terms in equations 2.3 and 2.4 to obtain the corresponding linear system:

d

dz

(

Ps

Pp

)

=

(

−αs 0

0 −αp

)(

Ps

Pp

)

(3.27)

The eigenvalues and eigenvectors of 3.27 are respectively:

r1 = −αs, ξ(1) =

(

1

0

)

(3.28)

r2 = −αp, ξ(2) =

(

0

1

)

(3.29)

The solution is then given by the following equation:

46

3.4. Stability analysis

(

Ps

Pp

)

= C1

(

1

0

)

exp (−αsz) + C2

(

0

1

)

exp (−αpz) (3.30)

where the constants C1 and C2 are computed with the initial conditions.

The eigenvalues are both real and negative, so, by looking at Table 3.1 we conclude

that the origin of the phase plane is a node asymptotically stable. It follows from

equation 3.30 that when z → ∞, (Ps, Pp) → 0 regardless of the value of C1 or C2. In

other words, all solutions approach the critical point as z → ∞. If the solution starts

on the line ξ(1) then C2 = 0. Consequently the solution remains in the line ξ(1) for all z

and approaches the origin as z → ∞.

Returning to the nonlinear system 2.3 and 2.4, by dividing equation 2.4 by equation

2.3, we obtain the following:

dPp

dPs

=−(

αp +νpνsgR

)

Pp

(−αs + gR)Ps

(3.31)

Equation 3.31 is separable and can be written in the following way:

−αs + gRPp

Pp

dPp =−αp +

νpνsgRPs

Ps

dPs (3.32)

By integrating implicitly, we obtain the equation bellow:

νpνsgRPs + αp lnPs − αs lnPp = C (3.33)

where C is a constant of integration.

The representation of the curves assigned by equation 3.33 in the phase plane is

designated by phase portrait of the system. For pump and signal wavelengths at

1445 nm and 1562 nm, the phase portrait is depicted in Figure 3.2. We plot only the

first quadrant because only this part of the phase portrait is physically meaningful.

3.4.1 Lyapunov second method

The concept of Lyapunov stability is a broader method that requires no knowledge

of the solution of the system of 3.24. The conclusions about the stability are obtained

by constructing a suitable auxiliary function [23]. The technique is very powerful and

provides a more global type of information being valid for equations that are definitely

nonlinear.

Basically, Lyapunov second method is a generalization of two physical principles for

conservative systems, (i) a rest position is stable if the potential energy is a local

47


Data signal power (W)

Pum

p si

gnal

pow

er (

W)

1 2 3 4 5 6 7 8 9

x 10−3

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 3.2: Phase portrait.

minimum, otherwise it is unstable, and (ii) the total energy is constant during any

motion. Hence, to use Lyapunov second method to analyze the stability of the

origin, a positive definite auxiliary function V that contains the critical point has to be

constructed. This function has to be continuous, with continuous partial derivatives.

If the derivative of that function is negative definite, then the origin is a asymptotically

stable point.

For our problem, in the one forward pump one signal situation, V was defined

as [22]:

V (Pp, Ps) = aP 2p + bPpPs + cP 2

s (3.34)

where a > 0 and 4ac > b2, with Pp, Ps > 0.

Since the partial derivatives in respect to P1 and P2 are continuous and V is positive

definite for the domain of interest, we only need to study the signal of its spatial

derivative. Thus:

dV

dz= (2aPp + bPs) (−αpPp + gRPpPs) + (2cPs + bPp)

(

−αsPs −νpνsgRPpPs

)

(3.35)

The derivative of V is negative definite if c > aνs/νp.

3.4.2 Second equilibrium point

For a general Raman amplifier with m waves, to obtain the second equilibrium

point, it is necessary to solve the equation below:

− αi + AijPj = 0 (3.36)

48

3.5. Average Power Analysis

where Aij is given by:

Aij =

0 ⇐= i = j

− νiνjgij ⇐= i < j

gij ⇐= i > j

(3.37)

For waves with equally spaced frequencies ∆ and simplifying g by a linear function

of ∆ν from 0 to 13 THz and zero elsewhere, matrix A is given by:

Aij =

0 ⇐= i = j

−(

1− νi∆

)

(j − i) gmax

∆νmax⇐= i < j

(j − i) gmax

∆νmax⇐= i > j

(3.38)

For a generic number of equations m, it is not possible to find an expression for the

inverse matrix A−1, so the inverse must be computed for each situation using Gauss

Jordan elimination or LU decomposition among other methods [24].

3.5 Average Power Analysis

The Raman propagation equations are also solvable through semi-analytical

methods, using the average power analysis (APA) presented by Min et al [25]. This

technique that proved to be successful in the simulation of multi-signal EDFAs

[26] consists on the implementation of an analytical solution of the system over an

infinitesimal length in which there is no powers dependency with position. Thus, the

amplifier is splited into n small segments, in order to avoid the position dependency

of the powers of equations 3.4, 3.5 which are then solved analytically in each segment,

considering as input conditions the outputs provided by the solution on the previous

segment. Equations 3.39 to 3.43 show how the powers are iteratively computed. The

output pump/signal power at each section end is given by:

P±out = P±

inG (z, ν) (3.39)

being G (z, ν) the section gain, as stated below.

G (z, ν) = exp [(−αν + Aν −Bν)∆z] (3.40)

The constants, Aν and Bν are obtained through:

Aν =i−1∑

j=1

gijPj (3.41)

Bν =m∑

j=i+1

gijPj (3.42)

49


The optical power term in each section can be substituted by its length averaged

values given by:

P = P±in

Gν − 1

lnGν

(3.43)

z=0 z=Ldz

Fiber length

Ps(z=0) known

Pp(z=0)unknown

Ps(z=L)unknown

Pp(z=L)unknown

1 2 3 n... n-1n-2

Pin PoutG

Figure 3.3: Sheme of the implemented method for one data signal and one backward pump.

1450 1500 1550 1600 1650 1700 1750 1800

0,0

2,0x10-4

4,0x10-4

6,0x10-4

8,0x10-4

Ram

an g

ain

effic

ienc

y (W

-1m

-1)

wavelength (nm)

Figure 3.4: Raman gain efficiency spectrum. The points were obtained experimentallyaccording to [27].

The numerical issues due to the backward pumping can be surpassed by assuming

that the pump inputs are located at the same fiber end that the signal inputs.

Therefore, the pump equations are integrated reversely as if they were backward

by multiplying them by (-1). A guessed initial input is necessary to perform the

integration, but the algorithm is able to adjust it using an optimization routine that

50

3.5. Average Power Analysis

adjust the initial input until the output at the fiber end reaches the real backward

pump input. A scheme of the procedure for the simpler situation of one data signal

and one backward pump is depecited in Figure 3.3.

To demonstrate the numerical resolution of the steady-state Raman propagation

equations, we assume the scheme in Figure 2.4, where three bidirectional pumps

(two backward and one forward) and four probe signals are considered. The counter

propagated pumps have power levels set equal to 0.1 W, working at 1450 nm and

1460 nm, respectively. The co propagated pump is working at 1470 nm with an output

power also equal to 0.1 W. The forward pumping signal are then injected into 40 km

of SSMF fiber and combined with 4 × 1000 GHz spaced C band probe signals with

an initial optical power equal to 1 µW . The simulation has considered attenuation

coefficients of 0.22 dB/km and 0.25 dB/km for data signals and pumps, respectively.

The Raman gain efficiencies are the ones depicted in Figure 3.4 where an effective

are of 80 µm2) was considered. The convergence of solutions using a spatial step of

500 m was assessed previously and thus used in the APA implementation. The spatial

evolution of pumps and probe signals are displayed in Figures 3.5 and 3.6.

0 10 20 30 400.6

0.7

0.8

0.9

1.0

1.1

1.2

Pro

be P

ower

(W

)

Fiber length (km)

1530 nm 1538 nm 1546 nm 1554 nm

Figure 3.5: Spatial evolution of 2 counter propagated pumps, 1 co propagate pump and 4 probesignal along a 40 km SSMF fiber span amplifier. Probe signals evolution.

Although the backward or bidirectional pumping schemes require the use

optimization routines to determine the conditions at z = 0 that could potentially

increase the computational effort of the implementation, it remains efficient because a

semi-analytical method is used to implement Raman equations. Indeed, the use of the

APA approach has shown a reduction of two orders of magnitude in the computation

51


time, being the obtained results in agreement with the ones resulting from traditional

numerical methods. In chapter 4, where optimization issues are studied, it will be

demonstrated that APA is a very reliable method to accomplish a Raman amplifier

simulator.

0 10 20 30 400.00

0.02

0.04

0.06

0.08

0.10 1450 nm 1460 nm 1470 nm

Pum

p P

ower

(W)

Fiber length (km)

Figure 3.6: Spatial evolution of 2 counter propagated pumps, 1 co propagate pump and 4 probesignal along a 40 km SSMF fiber span amplifier. Pump signals evolution.

3.6 Experimental validation

To validate the results obtained from the numerical methods, namely collocation

and APA, the experiments described in [28], [29] were carried .

Probe signals

Pumping lasers

Fiber

WDM/coupler

Isolator

OSA

Figure 3.7: Scheme of the experimental setup.

52

3.6. Experimental validation

1400 1420 1440 1460 1480 1500

-40

-30

-20

-10

0

10

Opt

ical

Pow

er (d

Bm

)

Wavelength (nm)

Figure 3.8: Measured optical spectrum for the 4-pump Raman module at 20 percent of the totalpower (A 3 dB attenuator was considered in this measurement).

1530 1540 1550 1560 1570

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

On/

off g

ain

(dB

)

Wavelength (nm)

simulation meas

Figure 3.9: Measured and simulated on/off gain.

The setup is composed by a tuneable laser to generate the probe signals (1530,

1540, 1550, 1560 and 1570 nm) and a backward pumping module (Amonics ARA-

53


CL-4B-R) with 4 diode lasers at 1426, 1444, 1462 and 1487 nm. The propagating

medium is 40 km of SSMF. A passive coupler is used to assemble the probe and

pumps signals and an isolator was used before the fiber to protect the signal source,

according to the scheme depicted in Figure 3.7. The used fiber Raman gain coefficient

was based in previously published results [27]. We set the signal initial power equal

to 3 dBm and measured the fiber input and output powers at each wavelength with

a power meter. The signal power was firstly assessed with the pumping module off

and then with the module at 20 percent of the total power. Afterward, the on/off

gain was computed. Figure 3.8 depicts the spectrum of the 4-pump Raman module

at 20 percent of total power (corresponding to the 78.14 mW, 55.76 mW, 15.80 mW

and 59.20 mW for the pump signals), measured with an Optical Spectrum Analyser

(Advantest), with a resolution of 0.1 nm. This configuration for the pumps optical

powers produces the gain spectrum depicted in Figure 3.9. The reduced chi-square

between the experimental and simulation data attained a value of 0.0655 which

demonstrates that the experimental and simulation results are in good agreement.

Further indirect experimental validations of the simulation were carried on, namely

to study the enlargement and optimization of gain excamined in chapter 4.

3.7 Chapter summary

This chapter described the work that was carried out concerning the modeling of

Raman Fiber amplifiers. It starts with an introduction that sets the main mathematical

challenges on the implementation of Raman propagation equations, namely the

intrinsic difficulties of nonlinear ordinary differential equations defined by boundary

value problems. A mathematical model for multi-pump and multi-signal RFA is

then introduced, given by the time and space evolution of pumps and signals,

being a steady state model presented afterwards. The practical issues related to

the numerical implementation of the steady state model are discussed and some

methods are introduced such as shooting and collocation. Improvements to the

traditional shooting are then discussed in terms of accuracy and efficiency, namely

an enhanced Runge-Kutta method proposed by Xueming Liu. The numerical analysis

proceeds with the explaination of collocation method and its practical implementation

using the Matlab solver, bvp4c is also explained and discussed. After the survey on

numerical methods, some analytical approaches are presented. Important insigths

are provided by stability analysis and Lyapunov second method. In the scope of

Raman propagation equations, we conclude that the only critical point with physical

significance is the origin and that it is a node, asymptotically stable. The trajectories

of the solutions in the phase plane demonstrate this conclusion. After that, efficient

methods based on semi-analytical integration, such as APA are presented and their

accuracy compared with pure numerical methods, as collocation.

54

3.7. References

References

[1] A. A. Tio and P. Shum, “Gain properties of multi-

wavelength time division multiplexed raman amplifier,” Opt. Express,

vol. 14, no. 12, pp. 5061–5066, 2006. [Online]. Available:


[2] M. Tang, P. Shum, and Y. Gong, “Design of double-pass discrete

raman amplifier and the impairments induced by rayleigh backscattering,”

Opt. Express, vol. 11, no. 16, pp. 1887–1893, 2003. [Online]. Available:


[3] I. Model, “Time-domain simulation of power transients in Raman fibre

amplifiers,” Int. J. Numer. Model, vol. 17, pp. 165–176, 2004.







[6] V. Perlin and H. Winful, “Optimal design of flat-gain wide-band fiber Raman

amplifiers,” Journal of lightwave technology, vol. 20, no. 2, p. 250, 2002.

[7] W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical recipes in

FORTRAN: the art of scientific computing. Cambridge Univ Pr, 1992.

[8] X. Liu, “Fast and accurate algorithms for solving model equations of fibre

amplifiers,” Journal of Optics A: Pure and Applied Optics, vol. 6, p. 961, 2004.

[9] X. Liu and B. Lee, “Effective shooting algorithm and its application to fiber

amplifiers,” Opt. Express, vol. 11, no. 12, pp. 1452–1461, 2003. [Online]. Available:


[10] U. Ascher, R. Mattheij, and R. Russell, Numerical solution of boundary value problems

for ordinary differential equations. Society for Industrial Mathematics, 1995.

[11] L. Shampine, Numerical solution of ordinary differential equations. Chapman &

Hall/CRC, 1994.

[12] A. Karim and I. Abbas, “The stability of the fourth order Runge-Kutta method

for the solution of systems of differential equations,” Communications of the ACM,

vol. 9, no. 2, pp. 113–116, 1966.

55





[13] Q. Han, J. Ning, H. Zhang, and Z. Chen, “Novel shooting algorithm for highly

efficient analysis of fiber raman amplifiers,” Lightwave Technology, Journal of,

vol. 24, no. 4, pp. 1946 – 1952, april 2006.

[14] J. Stoer and R. Bulirsch, Introduction to numerical analysis. Springer Verlag, 2002.

[15] G. Hall and J. Watt, Modern numerical methods for ordinary differential equations.

Oxford University Press, USA, 1976.

[16] J. Kierzenka and L. Shampine, “A BVP solver based on residual control and the

Maltab PSE,” ACM Transactions on Mathematical Software (TOMS), vol. 27, no. 3, p.

316, 2001.

[17] L. Shampine, J. Kierzenka, and M. Reichelt, “Solving boundary value problems

for ordinary differential equations in Matlab with bvp4c,” Manuscript, available at

ftp://ftp. mathworks. com/pub/doc/papers/bvp, 2000.

[18] L. Shampine, I. Gladwell, and S. Thompson, Solving odes with matlab. Cambridge

Univ Pr, 2003.

[19] W. Boyce and R. DiPrima, Elementary differential equations and boundary value

problems. Wiley New York, 2001.

[20] M. Gockenbach, Partial differential equations: analytical and numerical methods.

Society for Industrial Mathematics, 2002.

[21] E. Chauvet, J. Paullet, J. Previte, and Z. Walls, “A Lotka-Volterra three-species

food chain,” Mathematics magazine, vol. 75, no. 4, pp. 243–255, 2002.

[22] B. Neto, M. Rodrigues, E. Rocha, and P. Andre, “Stability analysis of raman

propagation equations,” in Transparent Optical Networks, 2009. ICTON ’09. 11th


[23] J. La Salle and S. Lefschetz, Stability by Liapunov’s direct method: with applications.

Academic Press New York, 1961.

[24] S. Lipschutz and M. Lipson, Schaum’s outlines: linear algebra. Schaum’s Outline

Series, 2008.

[25] B. Min, W. J. Lee, and N. Park, “Efficient formulation of raman amplifier

propagation equations with average power analysis,” Photonics Technology Letters,

IEEE, vol. 12, no. 11, pp. 1486 – 1488, nov 2000.

[26] T. Hodgkinson, “Improved average power analysis technique for erbium-doped

fiberamplifiers,” IEEE Photonics Technology Letters, vol. 4, no. 11, pp. 1273–1275,

1992.

56

3.7. References

[27] M. Fugihara and A. Pinto, “Low-cost Raman amplifier for CWDM systems,”


[28] B. Neto, C. Reis, A. Teixeira, P. Andre, and N. Wada, “Gain equalization technique

for raman amplification systems based on the hybrid optimization algorithm,”

in Microwave and Optoelectronics Conference (IMOC), 2009 SBMO/IEEE MTT-S

International, 3-6 2009, pp. 687 –689.

[29] D. Sperti, P. André, B. Neto, A. Rocha, A. Bononi, F. da Rocha, and




57

Chapter 4

Gain optimization of Raman amplifiers

4.1 Introduction

One of the most impressive features of Raman fiber amplifiers is assuredly the

possibility to achieve gain at any wavelength, by selecting the appropriate pump

wavelengths. By this way, it is possible to operate in spectral regions outside the

Erbium doped fiber amplifiers bands over a wide bandwidth (encompassing the S,

C and the L spectral transmission bands).The feasibility of this feature is assured

by assembling several pumps through WDM couplers into a single fiber to realize

a composite Raman gain. The composite Raman gain created by different pump

wavelengths can therefore be taylored in order to be arbitrarily broad and equalized.

The practical implementation of WDM-pumping requires that the pump lasers ought

to have narrow and stable lasing spectra. Typically, laser diodes with output powers

in the 100-200 mW range and operating in the 14XX nm region can be used in

a multipump scheme. Until the mid-nineties of the twentieth century, the costs

related to the deployment of those pumping lasers have been a barrier for the wide

implementation of multipump Raman amplification. However, numerous efforts and

the evolution of technology have allowed this field to evolve and nowadays high

power pumps are commercially available. Theoretically, the higher the number of

pumps, the better the gain equalization. Nevertheless there are economic and safety

issues that prohibit the use of an arbitrary number of pumps. Thus, a balance

between the system performance and the cost of amplification must be settled. The

practical implementaion of Coarse WDM based systems had also open the way for

the development of low cost Raman solutions because the latter are not demanding

in terms of wavelength stability [1]. Thus, alternative pumping techniques, such as

incoherent single pumping [2], [3], [4] [5], an array of low power pumps [6] or hiher

order Raman pumping are valuable [7], [8].

Raman amplification using incoherent single pumping is a technique that proved

to be sucessful on the purpose of providing flat gain at low cost. This pumping

method approximates the pump spectrum by a high number of pumps with narrow

spectral width, which in practise works as an array of pumps. In contrast to the

conventional coherent pumping source, the pumping power of incoherent source

spreads over a wide wavelength range from several nanometers to tens of nanometers,

4. Gain optimization of Raman amplifiers

being the phase and polarization completely random. Thus, the Four-wave mixing

generation by the interaction of pump-pump, pump-ASE noise and pump-signal is

significantly reduced. Another advantage of using incoherent pumping is that the

polarization multiplexer for pumping sources can be eliminated due to the random

polarization of pumping sources. In terms of gain equalization, the benefits of this

technique are similar to multi-pumping, but there are competitive advantages in terms

of power budget because expensive multiplexers with high insertion losses are no

longer needed. A better noise performance is also obtained at lower cost. For Raman

amplification using conventional coherent pumping, a bi-directional scheme is needed

to obtain simultaneously equalized gain and noise figure, as well as OSNR, however,

incoherent pump sources can improve the latter because the ASE peak can be shifted

outside of the transmission window [9].

The techniques that makes use of an array of low power WDM pumps for Raman

amplification has also demonstrated to be efficient, but the number of pumps to

achieve Raman gain threshold must be high and the additional insertion losses

resulting from multiplex them also need to be accounted for. However, this technique

has the merit of using affordable lasers that lasers that in practise produce better

performance in terms of gain enlargement then single pump. Published studies have

suggested the feasibility of this technique for the amplification of L-band transmission

channels using as pumps C-band lasers [6], among others assessments.

The multiple order Raman pumping is another interesting technique for gain

enlargement and equalization that makes use of a single pump. As incoherent

pumping, this technique needs no multiplexers, so the asociated insertion losses are no

longer a problem. The performance improvement provided by higher-order cascaded

counter-propagating Raman amplifiers arises from the fact that the final pump is

developd within the transmission span itself and, as a result, the incoming signals

are amplified earlier i.e. further from the receiver terminal. As shown in [10], the

optical signal-to-noise ratio increases as the order of the Raman pumping is increased,

at least for second- and third-order pumping. Bouteiller et al have shown the feasibility

of a dual order Raman pump for telecommunications using a single laser cavity. By

using this pumping technique, the performance in terms of noise figure is enhanced,

allowing flexibility for gain flattening [7]. Six order cascaded Raman pumping using

signgle Yb laser operating at 1091 nm was also demonstrated to be helpul in the

purpose of gain flattening and an improvement of 10 dB in the power budget was

presented [8].

The implementation of multipump Raman amplifiers presented several

challenging features, namely the strong Raman interaction between the pumps (pump

depletion). Since the higher frequency pump is responsible for the amplification

of the lower frequency signals, more pumping power is needed, as some will also

be transferred to the lower frequency pumps. This interaction between pumps also

60

4.2. Multi-pump allocation

affects the noise properties of the amplifier. However, some novel pumping schemes

have been recently proposed in order to avoid those unwanted effects: time dependent

Raman pumping, higher order pumping and broad-band pumping. Besides that,

multipump architecture requires that the pump operating parameters (powers and

wavelengths) must be properly settled in order to provide a defined level of gain.

This technique, so-called pump allocation, is usually based on powerful algorithms

that use time consuming heuristics.

The work presented in this chapter is mainly based on the implementaion of

an optimization tool in a way that is pretented to be simultaneously accurate and

efficient, followed by its subsequent experimental validation. A inovative heuristics

based on hybrid genetic algorithm (GA) is presented and the performance results

compared with simple GA. The results have demonstrated the feasibility of this

metodology in a pratical context because the needed simulation efforts are compatible

with the network reconfiguration times.

4.2 Multi-pump allocation

As said above the technique of multi-pump allocation consists on the

determination of initial powers and emitting wavelengths of the lasers pumps that

results on a flat gain around a pre-defined value. In such a metodology, the gain

enlargement is an imediate consequence of the addition of more pumps. The principle

beyond this technique is illustrated in Figure 4.1.

1400 1420 1440 1460 1480 1500 1520 1540 1560 1580 16000.20

0.22

0.24

0.26

0.28

0.30

0.32

0.34

Net gain (around 5 dB)

Gain contribution from each individual pump

pumps powers and wavelengths

Pum

p po

wer

(W)

Wavelength (nm)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Norm

alized gain coeffcient

Figure 4.1: Gain enlargement and equalization using multi-pump allocation.

61


It is noticeable that the asymmetry of the Raman gain efficiency spectrum denotes

that gain at the longest wavelengths is determined essentially by the location of the

longest wavelength pumps signals. On the other hand, the shorter wavelengths

receive gain contributions more uniformly from several pumps.

The gain equalization of channels in a small bandwidth by using a large number

of pumps is an easy tasks that a priori does not require the use or sophisticated

optimizations methods. On the other way, achieving a flat gain on a large bandwidth

( 80-100 nm) using a small number of pumps (typically bellow 12) is a very

cumbersome task which requires the use of metaheuristics [11], [12].

The use of those methods for optimization for the gain spectrum has been

widely performed, namely neural networks [13], simulated annealing [14] and genetic

algorithm (GA) [15]. This optimization problem has multi-variables (the pumping

parameters, usually twice the number of pumps) and can also be multi-objective

if additional conditions are settled. During the search process, the pump powers

and frequencies are directly substituted into the system of propagation equations

to calculate de gain profile. Hence, the optimization involve the resolution of a

highly nonlinear problem. Depending on the speed of the numerical method used

to integrate the system of equations, the amount of numerical computations involved

can be considerably large and the optimization inevitably time consuming. Those

solutions are not suitable for practical applications where the real optimal solution

must be provided in a short time.

4.3 Metaheuristics

In computer science, a heuristic is a technique designed to solve a problem that

ignores whether the solution can be proven to be correct, but which usually produces a

good solution or solves a simpler problem that contains or intersects with the solution

of the more complex problem [16]. An heuristic can be designated by metaheuristic if

the solving is based on the combination of back-box procedures, which are usually

heuristics too. Depending on the type of problem which needs to be solved, the

option for a metaheuristic should be cautious because the latter can be wasteful if

used indiscriminately and straitforward methods such as gradient based are much

more efficient.

The problem of gain optimization is typically suitable for metaheuristics because

it is multi-variable, highly nonlinear and presents constraints.

Genetic algorithms belong to the vaste body of evolutionary algorithms which

mimics the evolution of species [17]. In the context of the evolutionary algorithms,

the individuals that are subjected to evolution are the solutions, more or less efficient,

for a given problem. These solutions belong to the search space of the optimization

problem. If the problem is minimization one the best outcome is zero and the

62

4.3. Metaheuristics

proximity to that value is designated by fitness. The set of the individuals treated

simultaneously by the evolutionary algorithm constitutes a population. It evolves

during a succession of iterations called generations until a termination criterion, which

takes into account a priori the quality of the solutions obtained, is satisfied. During

each generation, a succession of operators is applied to the individuals of a population

to generate the new population for the next generation. When one or more individuals

are used by an operator, they are called the parents. The individuals originating from

the application of the operator are their offspring. Thus, when two operators are

applied successively, the offspring generated by one can become parents for the other.

The selection is the way to choose from the individuals of a population of possible

solutions the ones who will create offsprings for the next generations, and how many

offsprings each of them will create. The purpose of selection is to distinguish between

the fitter individuals in the population in hope that their offsprings will in turn have

even better fitness [17] [18].

The variation operators allow the algorithm to find better solutions than those

represented in the current population. The variation operators are able to transform a

given solution that was previously produced recurring to the selection operator. They

are classified into two categories:

1. mutation operators produce random change in an element of a string within the

given population;

2. crossover operators exchange the string fragments between two selected.

members of the population.

There are many ways of implementing the crossover, for example having a single

crossover point or multiple points which are randomly selected. A priori, it is

not possible to identify the most adequate crossover because its success and failure

depends on the particular fitness function, on the encoding and on other details [19].

The selection combined with the crossover gives the GA the bulk of its processing

power. The mutation is a random change in an element of a string within the given

population. As a matter of fact the mutation operator plays a secondary role in the

GA, being considered as a secondary mechanism of its adaptation. The working

principle of those operators, namely single point crossover and uniform mutation are

summarized in Figure 4.2. In single point crossover, the string is broken in two and

the two, being the resulting fragments recombined. In mutation a bit in the string can

randomly change its value.

In each generation, a generic evolutionary algorithm implements the loop iteration

summarized in Figure 4.3 that incorporates the application of the above mentioned

operators on the population:

1. for the reproduction, selection of the parents among a population of n

individuals to generate m offspring;

63


Offspring00110101

10101100

Crossover point

Parents

00110100

10101101

10101101 10101111

Crossover

Mutation

Figure 4.2: Crossover and mutation operation scheme.

2. crossover and mutation of the m selected individuals to generate m offspring;

3. fitness evaluation for the offspring;

4. selection for the survival of n individuals among the m offspring and n parents,

or only among the m offspring, according to the choice made by the user, in order

to build the population for the next generation.

Fitness evaluation of n individuals

m offspring + n parents

Crossover of the m selected individuals

m offspring + n parents

Mutation of the m selected individuals

n + mindividuals

Fitness evaluation of m offspring

Selection for the replacement (replacement)

STOP ?

n individuals

no

Best individual(s)

yes

Population initialization

n individuals

Selection for the reproduction (selection)

Figure 4.3: The generic evolutionay algorithm.

Although an intensive comprehention of evolutionary computation is beyond the

scope of this thesis, it is important to understand how the algorithm was implemented

64

4.3. Metaheuristics

to surpass the issues related to selective pressure. Indeed, there is a large amount of

work.

The performance of a genetic algorithm, like any global optimization algorithm,

depends on the mechanism for balancing the two conflicting objectives: exploiting the

best solutions and their ability to reproduce and at the same time exploring the search

space for promising solutions. In theory, the GA can combine both exploration and

exploitation at the same time if the following conditions are verified [20]:

1. the population is infinite;

2. the fitness function accuratey reflects the utility of a solution;

3. the genes in a chromossome do not interact significantly.

In practice, the population size is finite, which influences the sampling ability of a

genetic algorithm and as a result affects its performance. Due to its limited population

size, a genetic algorithm may also sample bad representatives of good search regions

and good representatives of bad regions. Thus, two important issues must be

accounted when dimensioning the selection operator: the maintenance of diversity

within the population and the selective pressure. These concepts can be understand

as the pression that individuals fitness exert on the overall evolution. The individuals

having the best fitnesses are reproduced more often than the others and replace

the worst ones. If the variation operators are inhibited, the best individual should

reproduce more quickly than the others, until its copies completely take over the

population. Thus the selective pressure will decide whether the algorithm produces

premature convergence or not. The stochastic nature of the selection operator also

introduce errors designated by genetic drift [21], [22].

There are several schemes for the selection process probability methods, scaling

techniques, tournament, elitist models and ranking models [17]. Probability methods,

such as roulette wheel, assign a probability of selection, Pj to each individual j based

on its fitness value. Thus, a serie of N random numbers of individuals is generated

and compared against the cumulative probability of the population, Ci =∑i

j=1 Pj . In

the particular case of roulette wheel the propability of selection is given by:

Pi =Fi

∑PopSizej=1 Fj

(4.1)

Ranking methods assign Pi based on the rank of a solution i when all solutions are

sorted. In the case of normalized ranking, the probability of selection is given by [23]:

Pi = q′ (1− q)r−1 (4.2)

65


where q is the probability of selection the best individual, r, the rank of the individuals

considering that 1 is the best and q’ is:

q′ =q

1− (1− q)PopSize(4.3)

Tournament selects j individuals randomly, with replacement from the population

and inserts the best of the j into the new population, repeating the process until N

individuals are selected.

In stochastic universal selection, we considers a straight line segment partitioned

in as many zones as there are individuals in the population, each zone having its size

proportional to the fitness. However, the selected individuals are designated by a

set of equidistant points, their number being equal to the number of offsprings. This

method is different from the proportional methods as here only one random drawing

is required to place the origin of the series of equidistant points and thus to generate

all the offspring in the population. As a results, the variance of the process being

weaker than in the roulette wheel method, the genetic drift is smaller. The issue of

genetic drift was considered as the dominant one in the optimization of Raman gain

spectrum.

A more deterministic selection option is Remainder, which performs two steps. In

the first step, the function selects parents deterministically according to the integer

part of the scaled value for each individual. For example, if an individual’s scaled

value is 2.3, the function selects that individual twice as a parent. In the second

step, the selection function selects additional parents using the fractional parts of

the scaled values, as in stochastic uniform selection. The function lays out a line in

sections, whose lengths are proportional to the fractional part of the scaled value of

the individuals, and moves along the line in equal steps to select the parents. Note that

if the fractional parts of the scaled values all equal 0, as can occur using Top scaling,

the selection is entirely deterministic [24] .

Incorporating a local search method within a genetic algorithm can help to

overcome most of the obstacles that arise as a result of finite population sizes.

Although genetic algorithms can rapidly locate the region in which the global

optimum exists, they take a relatively long time to locate the exact local optimum

in the region of convergence [25]. A combination of a genetic algorithm and a local

search method can speed up the search to locate the exact global optimum. In such

a hybrid methodology, applying a local search to the solutions that are guided by

a genetic algorithm to the most promising region can accelerate convergence to the

global optimum. The time needed to reach the global optimum can be further reduced

if local search methods and local knowledge are used to accelerate locating the most

promising search region in addition to locating the global optimum starting within its

basin of attraction.

66

4.4. Hybrid Genetic Algorithm

To enhance the efficiency of the GA, without any expense of accuracy, some hybrid

approaches have been proposed. A combination of GA and simulated annealing is

employed in [26] to optimize the power levels of 20 equally spaced pumps. GA is

implemented and a simulated annealing step is inserted after mutation to renew the

population of solutions and avoid premature convergence. Despite the flattening of

the spectral gain was achieved over a 100 nm bandwidth, the obtained ripple remains

relatively high (0.74 dB). Hybrid GA based upon clustering, sharing, crowding and

adaptative probability is used in [27]. This methodology explores the possibility of

finding multiple optima instead of only one by forcing the GA to maintain a diverse

population throughout its search. Therefore, several pumps schemes were tested

(with 2, 3, 4, 5, 6 and seven pumps) having been demonstrated that the improvements

in flattening the gain decrease the signal bandwidth.

4.4 Hybrid Genetic Algorithm

Considering the above reffered issues, the use of a hybrid GA (GA combined with

local search method) was the preferred methodology used in this thesis for the gain

optimization of broadband Raman amplifiers. Our approach is based on performing

GA until it reaches the optimal region instead of the optimal solution itself. The

remaining search can be done with the local search method. The question is to

decide the right moment to switch from one method to the other so that the algorithm

improves its optimization efficiency without lack of accuracy.

The Nelder-Mead method attempts to minimize a scalar-valued nonlinear function

of N real variables using only function values, without any derivative information

(explicit or implicit). The Nelder-Mead method thus falls in the general class of direct

search methods [28].

The method uses the concept of a simplex, which is a special polytope of N + 1

vertices in N dimensions. Examples of simplices include a line segment on a line, a

triangle on a plane, a tetrahedron in three-dimensional space and so on.

In the Matlab environment the Nelder-Mead is implemented in the optimization

toolbox using the function fminsearch. The algorithm first makes a simplex around the

initial guess x0 by adding 5 percent of each component x0(i) to x0, and using these N

vectors as elements of the simplex in addition to x0. Then, the algorithm modifies the

simplex repeatedly according to the following procedure.

1. Let x(i) denote the list of points in the current simplex, i = 1, ..., N + 1.

2. Order the points in the simplex from lowest function value f(x(1)) to highest

f(x(N + 1)). At each step in the iteration, the algorithm discards the current

worst point x(N +1), and accepts another point into the simplex. [Or, in the case

of step 7 below, it changes all N points with values above f(x(1))].

67


3. Generate the reflected point:

r = 2m− x(N + 1)

where

m =∑ x(i)

N, i = 1, ..., N

and calculate f(r)

4. If f(x(1)) ≤ f(r) < f(x(n)), accept r and terminate this iteration. Reflect

5. If f(r) < f(x(1)), calculate the expansion point s

s = m+ 2 [m− x(N + 1)]

and calculate f(s).

• If f(s) < f(r), accept s and terminate the iteration. Expand

• Otherwise, accept r and terminate the iteration. Reflect

6. If f(r) ≤ x(N), perform a contraction between m and the better of x(N + 1) and

r:

• If f(r) < f(x(N + 1)) (i.e., r is better than x(N + 1)), calculate

c = m+(rm)

2

and calculate If f(c) < f(r), accept c and terminate the iteration. Contract

outside.

Otherwise, continue with Step 7 (Shrink).

• If f(r) ≥ f(x(N + 1),

cc = m+(x(n+ 1)m)

2

and calculate f(cc). If f(cc) < f(x(n + 1)), accept cc and terminate the

iteration. Contract inside. Otherwise, continue with Step 7 (Shrink).

7. Calculate the N points:

v(i) = x(1) +(x(i)x(1))

2

and calculate f(v(i)), i = 2, ..., N + 1.

The simplex at the next iteration is x(1), v(2), ..., v(N + 1) Shrink

The scheme depicted in Figure 4.4 shows the points that fminsearch might calculate

in the procedure, along with each possible new simplex. The original simplex has a

bold outline. The iterations proceed until they meet a stopping criterion.

68

4.5. Simulation implementation

CC

s

r

c

m

v(N+1)

x(1)

x(N+1)

Figure 4.4: Scheme of the Nelder-Mead algorithm.

4.5 Simulation implementation

To enlighten the conclusions presented in the above section, we present the pratical

situation depicted in Figure 4.5. It is composed by 5 backward pumps injected into the

fiber by a multiplexer coupler and 16×400 GHz (from 1529.55 nm to 1577.86 nm) with

an initial power of 1 mW. Pumps and probe signals are then combined and injected

into 25 km of SSMF fiber. The signals were analyzed recurring to an optical spectrum

analyser (OSA) [29].

The simulation was carried out in a Matlab environment using the APA code

described in chapter 3 and the genetic algorithm toolbox [30] [23]. The toolbox enables

the implementation of different standardized operators for selection, crossover and

mutation. The fitness function was settled as the norm of the vector given by the

difference between each signal net gain and a target value. Additionally, a penalty

function was introduce to avoid the exploration of unwanted regions in the search

space, namely, negative valued powers and wavelengths.

The gain optimization simulation with this hybrid GA algorithm provided a mean

ripple below 0.17 dB. The optimized values for the pumps powers and wavelengths

are listed on Table 4.1.

4.5.1 Dimensioning of operators

Although the hybrid GA is always able to perform the gain optimization with a

ripple below 0.5 dB, some steps can be taken to improve the optimization in order

69


Probe signals

Optical Fiber

Coupler

Pumping lasers

OSAIsolator

Figure 4.5: Scheme of the implemented setup for the 5 pumps scenario.

Table 4.1: Optimized pumps wavelengths and powers.

Pumps Wavelengths (nm) Powers (mW)

1 1424.42 142.232 1433.28 89.583 1441.44 117.294 1464.71 187.545 1481.19 124.03

to reduce the number of function evaluations. The initial population where the GA

is achieved, as well as the GA operators, can be properly dimensioned so that the

algorithm is brought nearer the optimum in a faster way. Another important feature

is the number of generations needed to perform the GA before switching to the direct

search method. It is expectable that the increase in the number of generations provides

more chances to produce good fitted individuals enhancing in this way the process of

finding the minima. But the use of the hybrid configuration, i. e, the use of the GA

followed by a direct search method can avoid this necessity whenever the minima

reached by the GA are small enough.

The initial population range is the limit established for pump powers and

wavelengths at one end of the fiber. In the backward pumping scheme, the input

power values are not the real ones because the equation is integrated reversely. So,

it is advisable to try previously some values and to look to the output power values

at the other end of the fiber. Since the higher frequency pumps transfer more power

to the other waves it is expectable that at the end of the fiber length their powers are

lower. Therefore, it is advisable to try lower power values for the higher frequency

pumps and then increase them for the lower frequency pumps. Regarding the pumps

wavelengths, theoretical assumptions about Raman gain spectrum state that in order

to obtain a maximum gain, the signal beam is downshifted from the pump frequency

by 13.2 THz (about 100 nm in the 1500 nm region) [31] [32] [33]. So, it is advisable to

divide our spectral range into the number of pumps and then downshift those values

by 13.2 THz.

A good choice for the initial population range, i.e., a population range that

70


allows an exhaustive search from the algorithm is very important, because the next

generations are framed upon the best fitted members of the previous generation.

Based on the above mentioned statements and taking our five backward pumps

problem, the initial population for the powers are comprised between 1 mW and

10 mW for the first three higher frequency pumps. The powers for the other pumps

are settled between 10 mW and 100 mW. The initial population for the wavelengths

are [1430; 1440] nm, [1440; 1450] nm, [1450; 1460] nm; [1460; 1470] nm and [1470;

1480] nm, respectively.

In general it can be stated that the higher the number of individuals, the better the

fitness. Therefore, a better minimum will be found at the end of the run of the GA. The

inconvenient is that the running time can be too long because the simulation time is

proportional to the population size. In our scenario, we decided to perform the GA 10

times (since the random nature of GA will produce a different result at each run) and

to record the optimized gain ripples at the end of the first generation. Populations

with 25, 50, 75 and 100 individuals were tested. For each population, the mean of

the optimized ripples and the standard deviation were listed in Table 4.2. Although

the listed values suggest that higher sized populations present better minima, the

improvement is not worthwhile due to the increase in the simulation time. It should be

noted that exceptions to that rule sometimes occur on account of the above mentioned

random nature of this algorithm.

Table 4.2: Optimized ripples (means and standard deviations) obtained in 10 trials at the endof the first generation for population sized with 25, 50, 75 and 100 individuals.

Population size Mean value (dB) Standard deviation (dB)

25 4.176 0.59650 3.871 0.46475 3.607 0.449100 3.591 0.587

The considered GA selection methods are: stochastic uniform, roulette wheel,

tournament and remainder, which were explained section 4.3. For crossover we

can try different functions as heuristic, single point, two points, intermediate, and

scattered, which designates the way in which the individuals string is fragmented

and reconfigured. Heuristic crossover produces a linear extrapolation of the two

individuals taking into account their fitness values. The mutation can be performed

in two ways: uniform or gaussian. The uniform mutation selects one individual and

adds a random variable. A disadvantage of this mutation method is that it does not

prevent the solution of being trapped in a local minimum. Gaussian mutation adds

to an individual a Gaussian random variable N (0, σ), of zero average and standard

71


deviation σ, which has a probability density of:

f (y) =1

2πσexp

(

−1

2

(y

σ

)2)

(4.4)

Several computational experiments for the five backward pumps problem were

performed taking different kinds of selection methods: stochastic uniform, roulette

wheel, tournament and remainder. For each selection method, ten simulations were

carried out and the optimized ripples at the end of the GA were registered. Those

values are shown in Table 4.3. From the listed results (mean and standard deviation)

it is possible to conclude that the stochastic uniform method is the most appropriate

selection method for our purposes, being the tournament method the one that presents

the worse results.

Table 4.3: Optimized ripples (means and standard deviations) obtained in 10 trials for differenttypes of selection methods at the end of 10 generations.

Selection Mean value (dB) Standard deviation (dB)

Stochastic Uniform 2.139 0.435Roulette wheel 2.323 0.635Tournament 2.768 0.637Remainder 2.477 0.653

In order to find the best crossover function for our purposes, a GA with a

population size of 25 individuals was run along with 10 generations for the five

backward pumps problem. Among the available crossover functions, we concluded

that best one is the scattered crossover and the worse one is the intermediate crossover,

as displayed in Table 4.4 where the mean values of the optimized ripples for ten trials

are displayed as well as their standard deviation.

Table 4.4: Optimized ripples (means and standard deviations) obtained in 10 trials for differenttypes of crossover methods at the end of 10 generations.

Crossover Mean value (dB) Standard deviation (dB)

Scattered 2.139 0.435Two points 2.541 0.561Intermediate 2.809 0.501Single point 2.267 0.618Heuristic 2.566 0.775

Lastly, the same procedure was implemented for the mutation operator, being the

optimized ripples (mean and standard deviation) listed in Table 4.5. The best mutation

operator is uniform.

72


Table 4.5: Optimized ripples (means and standard deviations) obtained in 10 trials for differenttypes of mutation methods at the end of 10 generations.

Mutation Mean value (dB) Standard deviation (dB)

Uniform 2.139 0.435Gaussian 2.446 0.725

In conclusion the best GA operators are stochastic uniform for selection, scattered

for crossover and uniform for mutation.

4.5.2 Dimensioning hybrid GA

After checking the most adequate GA operators for the defined 5 backward pumps

problem, it is time to implement the hybrid algorithm (the GA followed by the

Nelder-Mead method). We said previously that this methodology was efficient and

accurate. To prove that the hybrid GA is more accurate then the simple GA, we have

implemented it and registered the optimized ripples of the simple GA and hybrid

GA with the number of generations. A population of 50 individuals was considered

and the adequate operators found in the section above were used. The Nelder-Mead

algorithm runs until the average change of the objective function, defined here as the

best attained gain ripple, is less than 1× 10−6.

In Figure 4.6, the GA and the hybrid GA minima are plotted for different number of

generations. It is clear that even when the algorithm is switched at a higher minimum

(the 5 generations situation) the Nelder-Mead method is able to reach a desirable

minimum. Hence, the hybrid method is assuredly more accurate then the simple GA.

It is also noticeable that the GA minima present exponential decay behaviour (r2 =

0.99549) with the number of generations and consequently present an asymptotic

behaviour as the number of generations increase unlimitedly. Thus, we can conclude

that it is not worthwhile the unlimited increase the number of generations in the hope

of improving the method accuracy. To improve the efficiency of the hybrid GA,

we have to identify which situation is the fastest or, alternatively, to identify the right

moment to switch from one method to another. By examining Figure 4.6, we realize

that the ripple variation performed by the Nelder-Mead method is higher for GA

with smaller number of generations (bellow 15), so it is expectable that more function

evaluations are needed and consequently more computational time.

Considering the 5 generations situation as a reference, we decided to plot the

relative computational time and the ripple variation attained by the Nelder-Mead

method against the number of generations and display the results in Figure 4.7. Since

both results decay when the number of generations is increased, the Nelder Mead time

is primarily dependent of the ripple value where it starts the search.

73


0 5 10 15 20 25 30 35 40 450,0

0,5

1,0

1,5

2,0

2,5

3,0

Bes

t rip

ple

(dB

)

Number of generations

GA GA+Nelder-Mead

Figure 4.6: Optimized ripples at the end of the GA and the hybrid GA for different number ofgenerations for a population size of 50 individuals. The line is the exponential interpolation.

0 5 10 15 20 25 30 35 40 45-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Number of Generations

Rip

ple

varia

tion

(dB

)

0.2

0.4

0.6

0.8

1.0

1.2

Relative N

elder-Mead tim

e

Figure 4.7: Relative Nelder-Mead time and the corresponding ripple variation. The lines arevisual guides.

74


0 5 10 15 20 25 30 35 40 45

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

t/t re

f

Number of generations

GA+ Nelder-Mead GA Nelder-Mead

Figure 4.8: The GA, the Nelder-Mead and the total simulation times for a hybrid GA againstthe number of generations for a population size of 50 individuals. The lines are visual guides.

In Figure 4.8, we plotted the GA, the Nelder-Mead and the GA + Nelder-Mead

relative simulation times against the number of generations. The reference is the

time necessary to accomplish an optimization using 40 generations in the GA and

continued by the Nelder-Mead. Here, we took the 40 generations situation as a

reference. We noticed that the computation time of the simple GA increases linearly

with the number of generations and that the tendency of the Nelder-Mead simulation

time is to decay. An optimal situation is attained when both the GA and the Nelder-

Mead achieve their computation taking a similar amount of time; thus, we realize that

this situation is verified when 17 generations are used in the GA.

4.6 Experimental validation

The work reported in this section describes the experimental assessment of the use

of multipump architecture instead of single pump and also validates the optimization

algorithm described above. The work was carried out in the optical communications

laboratory of the Instituto de Telecomunicações (IT) cited in Aveiro and in the facilities

of the National Institute of Communications and Information Technology cited in

Tokyo - Japan, in colaboaration with the Photonic Network group.

An important feature to properly design an amplification scheme is the evaluation

of the information channels bandwidth that is needed and the channel spacing. In

CWDM, both bandwidth and channel spacing are large, therefore, we have to chose

the best pumping scheme that produces a flat spectrum gain as large as possible.

75


To demonstrate a proper pumping scheme (one, two or three pumps), some

experiments were carried out [1]. A MUX with a set of three CW lasers was used

as a pumping unit, being either one, two or three lasers used in each experiment.

The copropagating pumps are centred at 1470 nm, 1490 nm and 1509 nm, each with

an output power of 22 dBm. A span of 40 km of SSMF with α = 0.22 dB/km and

Aeff = 80 µm2 is the propagation medium. The experiments were carried out with

the measurement of the system performance (on/off gain and noise figure) for each

scheme using a forward pumping configuration. The scheme depicted in Figure 4.9

reproduces the implemented experimental setup.

MUX

P1 P2 P3

Tx Rx

Transmitter Receiver

Fiber

Coupler

Figure 4.9: Scheme of the used experimental setup.

A comb of 12 wavelengths is used, but only three are turned on in each

measurement. The three active channels are equally spaced by 20 nm (the CWDM

channel spacing), starting with the triplet (1530, 1550, 1570 nm), then rigidly shifting

the three wavelength by 10 nm. Usually, this kind of measurements is taken with

only one active channel but measurements using three active channels simultaneously

instead of only one provide a more accurate assessment of the system performance.

For a single pump scheme, both simulation and experimental gain results are

displayed in Figure 4.10.

We observe that the results comply with the theoretical statement that the gain

peak is obtained for a wavelength upshifted from the pump by 100 nm [32]. This peak

decays rapidly with the wavelength, decreasing 1 dB for a small wavelength variation.

A 1 dB decay is equivalent to a declining of 20 percent of the maximum gain, in linear

units, and used to define the effective gain bandwidth. Therefore, the effective average

gain bandwidth is equal to 39 nm, allowing the use of only two active channels for

CWDM purposes. It must be noted that all three pumps present the same behavior.

76


1520 1540 1560 1580 1600 1620 16400,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

5,0

Gain

On-O

ff (dB

)

Wavelength (nm)

Simulation Experimental

PUMP 1470 nm

1 dB

(a) 1470 nm pump

1520 1540 1560 1580 1600 1620 16400,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

5,0

1 dB

PUMP 1490 nm

Gain

On-O

ff (dB

)

Wavelength (nm)


(b) 1490 nm pump.

1520 1540 1560 1580 1600 1620 16400,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

5,0

1 dB

Gain

On-O

ff (dB

)

Wavelength (nm)


PUMP 1509 nm

(c) 1509 nm pump.

Figure 4.10: Gain spectrum for the single pump scheme. Only the experimental bandwidth isassigned in the graphs.

After assessing the system performance using a single pump, we perform the

evaluation for multipump schemes, using the three pumps simultaneously. The

77


obtained gain spectrum is plotted in Figure 4.11. As opposed to the single pump

case, the deployed gain is higher and broader; being the effective bandwidth equal

to 48 nm. An 78 nm bandwidth was measured at 3 dB. Since the channel spacing

in CWDM is 20 nm, this pumping configuration allows the allocation of three active

channels.

1520 1540 1560 1580 1600 1620 16400

1

2

3

4

5

6

7

8

9

10

11

Gai

n O

n-O

ff (d

B)

Wavelength (nm)


Multi pump

Figure 4.11: Gain spectrum for multipump scheme. Only the experimental bandwidth isassigned in the graphs.

Concerning optimization, two different experiment are described, one for C

band optimization [34] and another focused on broadband amplification. Although

the second experiment requires an intensive simulational work, boths tasks were

succeded using the hybrid GA.

This experiment was used to verify gain equalization for transmission channels in

the C-band. The experimental setup follows the scheme depicted in Figure 4.5 and

is composed by tuneable lasers to generate the test signals (1530, 1540, 1550, 1560

and 1570 nm) and a backward pumping module (Amonics ARA-CL-4B-R), with 4

diode lasers at 1426, 1444, 1462 and 1487 nm with a total pumping power around 1 W.

The measured spectrum of the module is available in Figure 3.8, where the powers

were set at at 20 percent of the total power and a 3 dB attenuator was used to protect

the OSA.The propagating medium is 40 km of SSMF with an average attenuation of

0.22 dB/km. A passive coupler is used to assemble the test and pumps signals, being

used an isolator before the fiber to protect the signal source. The used fiber Raman

78


gain coefficient was based in previously published results [35].

The test signals had an initial optical power of 3 dBm. The fiber output test signals

power was firstly assessed with the pumps off and then with the pumps at a given

percentage of their maximum power. Afterward, the on/off gain was computed from

these values. The optical power of each pump was optimized, through the hybrid

algorithm using the methodology described in the section above. Thus, we employ

a population of 50 individuals with following GA operators: stochastic uniform for

selection, scattered for crossover and uniform for mutation. The GA was evolved

over 17 generations and switched to the Nelder-Mead method. The objective function

was settled as the gain ripple around a target on/off gain of 4 dB. The obtained

experimental and simulation on/off gain are displayed in Figure 4.12. The reduced

chi-square between the experimental and simulation optimized data attained a value

of 0.054.

Analyzing the experimental and simulation results, we notice that they are in

good agreement (reduced chi-square value of 0.054). For the optimized situation,

the obtained experimental gain ripple was equal to 0.24 dB. This demonstrates that

the hybrid GA algorithm is suitable to perform optimization for the Raman gain in

reconfigurable optical networks.

1530 1540 1550 1560 15703.50

3.75

4.00

4.25

4.50

On/

off g

ain

(dB

)

Wavelength (nm)

simulated measured

Figure 4.12: Measured and simulated on/off gain for optimized pumping power.

As done with the C-band equalization, an experiment to equalize the gain of

channels in a wide bandwidth (around 80 nm) was carried out. Thus, a Raman

amplified system with 20 km of SMF fiber, 20 probe signals and 7 backward pumps

privided by the Raman module (Fitel HPU-CL12W14D-00077) was implemented.

79


Since the pump wavelengths are already settled, only the optimization of the power

levels is needed. The simulation used the hybrid GA explained above being the

GA operators chosen to make to algorithm more efficient. Hence, the stochastic

uniform method was used for selection, being the scattered uniform methods used for

crossover and mutation. A population of 50 individuals was randomly created within

an predifined interval and a number of generations equal to 35 were considered. The

spatial evolution of the pumps signals optimized values are displayed in Figure 4.13

jointly with the pump signals experimental values. In Figure 4.14, the optimized and

experimental on/off gain spectra are presented.

This is a good agreement between the optimization modeling and the experiment.

The maximum ripple attained by the optimization is 0.41 dB being the experimental

maximum ripple equal to 0.23 dB. The mean square error between simulation and

experimental results is equal to 0.0036. Indeed, a flat gain over a wide bandwidth

(aroung 80 nm) was attained, using seven pumps with a total input power equal to

453 mW.

0 5 10 15 200

20

40

60

80

100

120

140

160

1410.0 nm 1424.2 nm 1437.9 nm 1452.1 nm 1465.5 nm 1494.8 nm 1502.4 nm

Pum

p P

ower

(mW

)

Fiber Length (km)

Figure 4.13: Power evolution of optimized pumps along 20 km of SMF (lines). The geometricshapes stand for the used experimental values.

Depite the arguments against, such as the cost of pump multiplexers and the

detrimental pump depletion effect and nonlinear effects as Four-wave mixing (FWM),

the multipump allocation is still a good solution for the purpose of gain enlargement

and equalization mainly due to the flexibility on setting the pumps architecture.

Even incoherent pumping, that is a very affordable solution does not allow a

simultaneous gain enlargement and equalization as efficient as multipumping because

80

4.7. Chapter summary

1520 1530 1540 1550 1560 1570 1580 1590 1600 1610

7.5

7.6

7.7

7.8

7.9

8.0

On/

Off

Gai

n (d

B)

Wavelength (nm)

Figure 4.14: Experimental (arrows) and simulation (line) on/off spectral gain for the 20 probesignals and 7 counter propagated pumps, over 20 km of SMF fiber.

the operational pump frequencies are simetrically placed around a central frequency.

Thus, when as the gain is enlarged, it becomes less equalized.

As demonstrated in this chapter, the problem of finding the pump parameters,

traditionally very time consuming, has been surpassed. Efficient and accurate

numerical solutions are able to provide the proper pump parameters for a particular

situation enabling the pratical implementation of the simulated solutions. Thus, the

option for this methododoly among others is still vantageous.

4.7 Chapter summary

This chapter described the work that was carried out within the topic of Raman

gain equalization, namely the gain enlargement and equalization by multi-pump

allocation. The chapter starts with an introduction presenting the state of the art of this

research topic and some important published methodologies to achieve Raman gain

equalization, such as incoherent pumping, pumping by an array of low power lasers

and multiple order Raman pumping are reffered. The approach developed in this

thesis is then presented and theoretically framed. Since the work requires intensive

computational effort and the use of robust optimization, the option for metaheuristics

is justified and the genetic algorithm principle of operation is explained. Practical

aspects concerning its implementation are also discussed and contextualized in the

problem of Raman gain equalization, being introduced, for that purpose, a practical

81


simulation situation. In order to improve the efficiency of the method, a hybrid

approach that combines Genetic Algorithm with a local search method is presented

and compared with simple GA. The results demonstrate that the computation time

can be reduced by a factor of two. To enligthen the results obtained in simulation,

two experiments were carried out. The first one assesses the possibility of obtaining

gain enlargement for CWDM applications, the last two achieve gain equalization for

(i) C band signals, using a backward 4-pump Raman module and (ii) over a wide

bandwidth (80 nm) using 7 backward pumps. The practical implementation has

shown that the simulation times are compatible with the network reconfiguration.

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4.7. References



85

Chapter 5

Optical amplifiers in access networks

5.1 Introduction

The fiber-to-the-home/premise (FTTH/P) access solutions are

of high technological and economical interest since they can implement a structure

in the field with reduced Capital expenditures (CapEx) and operational expenditures

(OpEx) dealing with increasing network data traffic. Access networks operate in a

local scale being connected to the metropolitan and then to the core networks.

OLT

RN

RN

RN

ONU

ONU

ONU

Feeder network Distribution network

Figure 5.1: General scheme of a PON and its subnetworks: feeder and distribution.

These so-called passive optical networks (PONs) consist of three fundamental

elements: the optical line terminal (OLT) at the service provider’s central office (CO),

the remote node (RN) and the optical network units (ONUs) in the end users. The

OLT is the equipment that enables the user services and controls the quality-of-service

(QoS) among other tasks. The OLT also carries out the multiplexing of users in the

optical fiber. The ONU converts the optical signal ariving from the OLT into an

adequate format to the end user, such as asynchronous transfer mode (ATM), Ethernet,

internet protocol (IP), etc. Inside the access network, two sub-networks can be defined,

the feeder and the distribution networks. The first one tranports the traffic between the

OLT and the RNs, and the second from the RNs to the ONUs. By definition, no active

elements can be included inside the distribution network. A general scheme of a PON

5. Optical amplifiers in access networks

and its elements is displayed in Figure 5.1. In the access network, two traffic directions

are allowed, one coming from the service provider to the end user, the down-stream

(DS), and another flowing in the opposite direction, the up-stream (US). The latter can

be assigned with equal of different wavelengths. If the same wavelength is used for

both DS and US, a pair of fiber is needed [1], otherwise, they can be multiplexed into

a single fiber, but their wavelengths must be chosen on order to prevent impairments.

The optical splitter/combiner in the RN distributes the DS information from the OLT

among the ONUs and combines the US coming from the distribution network. Thus,

all the elements are interconned using a low cost structure.

The deployment of broadband solutions , boosted by the proliferation of new web

applications, going from traditional search to social networks and video streaming,

had undergone a huge development thanks to WDM technology. However, due to

the limitation of access networks, the backhaul traffic can suffer a bottleneck in the

access structure pressing out the necessity of standartizing networks that provide

broadband services: broadband PONs (BPONs) [2], ethernet PON (BPON) [3] and

gigabit PON (GPON) [4]). The latter are currently based on time division multiplexing

techniques (time division multiplexing - passive optical network (TDM-PON)) and

have significant advantages in terms of instalation and OpEx. For TDM-PON, a

passive power splitter is used as the remote terminal, neing the signals multiplexed in

the time domain.

A counter part technology is wavelength division multiplexing where each ONU

is served by a single wavelength. This technology allows to increase the network

capacity and simplify the management as all the connections are point-to-point.

Besides that they exhibit a range of inherent advantages such as protocol transparency

and QoS. In Figure 5.2 are presented the two different approaches to provide

FTTH/P: TDM-PON and wavelength division multiplexing - passive optical network

(WDM-PON).

Until recently, the deployment of WDM-PON was limited by the costs because

the latter require additional transceivers operating on new enhancement bands

and filters to separate the wavelengths in the system [6] [7]. However, enabling

technologies are expected to upgrade the current TDM-PON to WDM-PON

enabling a smooth migration from one technology to the other. In [8], a

migration strategy is suggested using centralized ligth sources and novel scheduling

algorithms to share tunable components. In alternative, hybrid architectures such

as wavelength division multiplexing/time division multiplexing - passive optical

network (WDM/TDM-PON) [9], [10]or CWDM/TDM-PON [11]are valuable but still

experimental. However, these approaches are being pointed out as a possible solution

for the deployment for next generation access (NGA) networks and in FTTH/P

segment.

The maximum reach and split of a PON are determined by both the PON protocol

88

5.1. Introduction

Figure 5.2: Two different PON architectures: (left) TDM-PON; (rigth) WDM-PON [5].

and the physical layer optical reach. Extending the reach of a PON allows the

consolidation of the access and metro networks, reducing the number of switching

nodes which results in cost savings. The concept of increasing the reach and splitting

of a PON by means of optical amplifiers, had become a topic of research interest and

reach extenders have been standartized recently by the ITU-T (G.984.6). Solutions

based on the deployment in the field of optical amplifiers are valuable because,

unlike electronic equipments, they allow transparency to traffic format, bit rate and

simple network management. However, reach extenders may not be cost effective

because they require the use of electrically powered units in the field containing

optical amplifiers denying in this way the intrinsic benefits of PON [12]. Thus, the

most attractive solution for network operators rely on achieving extended reach while

maintaining the passiveness outside plant of the PON structure.

Typically, due to the 28 dB loss budget for Class B+ [12], the GPON deployments

are 1:32 split with a reach up to 20 km. Similarly, GE-PON offers loss budget of

20 and 24 dB, and achieves a maximum distance of 10-20 km. However, extended

reach have been achieved by adding in line amplifiers to existent infrastruture, with

the above mentioned drawback that this extenders need supply points along the

feeder line. However, Raman amplification solutions have intrinsic benefits because

the transmission medium is also the amplification medium so no power supply is

needed along the line, being the pumps located at the CO. As a consequence, Raman

approaches vary essentially in the type of pumping architecture. In [13] extended

reach access is attained with single fiber, bidirectional backhaul link and distributed

Raman amplification, achieving 80 km long reach PON link with symmetric up and

downstream data rate of 10 Gb/s. Alternatively, dual pumping at 1239 nm and

1427 nm can be used at the CO to extend the reach of GPON be means of Raman

amplification in AllWave fiber at 2.5 Gb/s, enabling a system reach of 60 km and

89


a 1:64 split ratio total in an entirely passive outside plant [14]. Lee et al have

used a similar approach in a rural scenario GPON, attaining 60 km extended reach

and 1:32 plit ratio on a symmetric 2.5 Gb/s bidirectional transmission [15]. Raman

amplification can also be used on WDM-PON using backward and bidirectional

pumping schemes. Additonally, pump recycling techniques can be valuable. In [16]

the re-use of non-negligible residual Raman pump power as a pump for an Erbium

fiber-based upstream broadband ASE source allows for fully-centralized control of

light sources at optical network units (ONUs), while distributed Raman amplification

over a transmission link enables us to obtain lossless low power signal transmission.

In WDM/TDM-PON architecture, the combined use of Raman amplification and

EDFA or remotely pumped EDF in a hybrid amplification scheme can have significant

merits because it allows the C+L bandwidth enlargement and introduce simplicity by

achieving with one 1480 nm laser the simultaneous pumping of EDFA and Raman.

However,this architecture brings additional issues coming from the transients.

Power transients can be caused by a sudden channel addition or removal in the

WDM environment caused by either unintentional failures, or by deliberate network

reconfigurations [17]. Thus, changes in the number of WDM channels will affect

the power of the surviving channels in a timescale proportional to the number of

amplifiers in the network [18]. During the transience, if the signal power increases,

the enhanced nonlinearities may distort the signal, while decreased signal power may

lead to OSNR impairment; and both effects will degrade the system performance [19].

Additionally, burst mode traffic can also originate transients. The main operational

concern with is effect is its duration and the amount of gain excursion.

It has been demonstrated that the effect of transients are more detrimental when

EDFA are used, because they are related with the population dynamics of the dopant

ions, and can be much faster than the ions relaxation time, depending on the signals

saturating power. Transients are also observed in RFA due to the dynamic pump-to-

pump and signal-to-signal interactions in the above described network invironment.

However, the transient duration and power surges is more noticeable in forward

multipump architectures. Surprisingly, transients have also been experimentally

observed even in counter-pumped single-channel saturated Raman amplifiers [20],

[21], [22]. The reason is that the strong power of the signal leading edge depletes the

injected pump, and thus the main body of the signal pulse does not enjoy the same

high gain as the signal front.

The work reported in this chapter addresses the use of optical amplifiers

in PON, namely RFA and EDFA (or remotely pumped EDF). The specific

architecture of SARDANA which is a hybrid WDM/TDM-PON is analyzed more

thoroughly concerning important operational issues such as the transients due

to the addition/dropping of channels and the ones owing to bursty/packeted

traffic. Concerning the EDFA or remotely pumped EDFA transients, mitigations

90

5.2. Transients in Raman amplifiers

strategies based on optical gain clamping were proposed and implememented in a

laboratorial context. Besides that, challenges concerning the bandwidth enlargement

covering C+L bands and the gain equalization were also projected recurring to

hybrid amplification schemes. Hence, an optimization strategy was developed by

introducing and enhanced order to add/drop channels and optimized EDF lengths.

5.2 Transients in Raman amplifiers

As said in the introduction section, the use of optical amplifiers, namely RFA for

burst-mode transmission, can originate transients. Although they are expected to be

more detrimental in saturated amplifiers such as EDFA, their occurence in RFA was

also demonstrated.

In RFA, the phenomenon of stimulated Raman scattering occur in a timescale

compliant with Raman time constant, TR, which assumes values in the fentoseconds

order for Silica based fibers. As approached in chapter 2, Raman scattering can

physically be interpreted by considering a short high power temporal optical pulse

incident on a Silica molecule. As a consequence, the molecule electrical polarization

vibrates with a very short period. Because this vibration period is very small when

compared with the duration of the pulse, molecular vibrations are induced. The

second term of the third-order nonlinear susceptibility, χ(3), is used to account the

molecular vibration and to model Raman effects [23], [24]. Thus, besides Raman gain

coefficient, γR, which is given by equation 2.1 the Raman time constant, TR, is related

to the slope of the Raman gain by [25]:

TR =2π

nn2

[

d(

Im(

χ(3)))

d (∆ω)

]

∆ω=0

(5.1)

Unlike EDFA which exhibit relaxation times in the order of the tens of milisecons,

the Silica medium responds almost instantaneously by stimulated Raman scattering.

However, when a suden variation in the total power that propagates inside an

optical fiber is observed, a time delay due to the propagation of data and pump

signals through the kilometers long gain medium fiber will cause a relatively slow

response on the amplifier. This effect is usually designated by Raman transient. The

response will depend deeply in the degree of gain saturation of the amplifier. If

the amplifier is operated at a linear gain regime the transient change in the gain

will not be so large. The observation of transients is commonly observed in WDM

systems due to the temporal variation of cross gain modulation when data signals

are added/dropped [26], [27]. In that situation the so-called surviving channel(s)

will experience temporarily more (or less) available pump power over a time interval

defined by the propagation of the perturbation along the fiber. Typically, the effect

is more penalizing in backwardly pumped systems [28]. Transience can be observed

91


for single data signal transmission backwardly amplified in a saturated amplification

regime [22]. The reason is that the strong power of the signal leading edge depletes

the opposite traversing pump, and thus the main body of the signal pulse does not

enjoy the same high gain as the signal front.

The computation of Raman transients are possible numerically by solving the

system of equations 3.1. The procedure starts by solving the system in steady state

conditions, using the methods described in chapter 3 and perform time evolution of

pumps, signals, and ASE waves. It must be noted that the time interval, ∆t, and

the space interval, ∆z, are connected by the group velocity, Vi, the perturbations

will propagate along the optical fiber. For backwardly pumped amplifiers, the

perturbation follows a retardated time frame ti = t− z/Vi. An analytical computation

of the gain excursion and transient time constant is also possible and accurate for

backwardy pumped Raman amplifiers. [22]. This approach is based on the derivation

of a relative integrated pump variation, measured in the signals retarded time frame.

For burst/packeted switching mode signal propagation, a perturbation of the

standing wave operated by an incoming packet at the frequency νi creates a notch

in the pump power that propagates backwards to the input. The reaction time of the

standing wave is at least the propagation time inside the amplifier. Hence, by the time

the standing wave reacts to the signal power variation, the notch has propagated to

the input, and thus we have an undampened power excursion across the signal output

pulse. This outcome is only possible for saturated Raman amplifiers.

To assess the impact of the use of burst traffic under Raman amplification, an

experiment was carried out. The underlying idea was to use different traffic features,

burst lengths and idle times, analyze the system performance and compare it with

EDFA.

The implemented experimental setup consists of an external cavity laser peaking

at 1550 nm, followed by a polarization controller and a Mach-Zehnder external

modulator driven at 10 Gbit/s. After passing through an optical isolator the optical

signal is injected into 45 km of SSMF. As a receiver, a NEL photonic packet, an Agilent

digital oscilloscope and an Anritsu BER tester were used.

To achieve Raman counterpropagation amplification we use a Fitel HPU-CL12W

pump set, containing 14 pump lasers, at the following wavelengths: 1410.0, 1410.2,

1416.9, 1417.3, 1424.2, 1429.8, 1437.9, 1444.3, 1452.1, 1458.1, 1465.5, 1472.6, 1494.8, and

1502.4 nm. The latter were driven with 500 mA, representing an aggregate optical

power of 1029 mW. The EDFA amplification scheme was implemented with one

device from IPG. In both cases, the optical power at the receiver was 0 dBm.

The tests were performed at 10 Gbit/s with 223 − 1 PRBS packet signals. For these

tests a composite signal was constructed, with packet sizes equal to 5000, 10,000,

15,000, and 20,000 bits, having all an off time equivalent to the previous burst size.

Table 5.1 displays the digital bit test sequence with a total length of 10 µs.

92

5.2. Transients in Raman amplifiers

Figure 5.3: Bit sequence at the receiver: top - Raman amplification, bottom- EDFAamplification. The vertical scale is arbitrary and the horizontal scale is 500 ns/div.

Table 5.1: Tested bit sequence.

5000 5000 10000 10000 15000 15000 20000 20000PRBS zeros PRBS zeros PRBS zeros PRBS zeros

The transient effect could be clearly noticed on the time domain, by observing the

bit sequence on the oscilloscope, for each amplification scheme, as shown in Figure 5.3.

Concerning the system performance, BER measurements were performed for the

two previously described amplification schemes, Raman and EDFAs, as well as the

back to back situation. The results are displayed in Figure 5.4. The difference between

the two configuration power penalties is 1.1 dB (for a BER at 10−11), being more

penalizing for the EDFAs amplification scheme. This could be mainly due to the ASE

noise and to transient effect associated with the EDFAs.

To access the network performance in the presence of packet traffic, we have

analyzed the Q factor at the receiver, as function of the inter-packet spacing in

93


-19 -18 -17 -16 -15 -141E-12

1E-11

1E-10

1E-9

1E-8

1E-7

B

ER

Receiver Optical Power (dBm)

Raman EDFA Back to back

Figure 5.4: Experimental BER for the Raman and EDFA amplification, and the back to backsituation. The lines are visual guides.

Figure 5.5: Packet signals bit sequences, the time between burst is 2 µs. Packet occupancydensities: from left to right and from top to bottom: 90, 70, 50, and 20 percent. The verticalscale is arbitrary and the horizontal scale is 500 ns/div.

the presence of Raman amplification with counterpropagation pumping. Figure 5.5

displays the test bit sequence for four distinct packets sizes, considering, in all cases,

time spacing between packets equivalent to 20000 bit. The packets size varies from

94

5.3. Raman amplification in hybrid WDM/TDM-PON

2000 to 18000 bits.

To understand the Raman amplification impact on the signal QoS, we compared

the Q factor in the back-to-back and in the transmission conditions. The results are

shown in Figure 5.6. On the same graph we display the difference between the two

curves, in order to analyze the packet occupancy density.

As predicted by the time constant of the Raman phenomenon, there is negligible

degradation on the Q as function of the packet density when the optical signal is

subject to Raman amplification scheme.

0.0 0.2 0.4 0.6 0.8 1.0

5.75

6.00

6.25

6.50

6.75

2.0 1.6 1.2 0.8 0.4 0.0

0.05

0.10

0.15

0.20 Transmission Back to back Diference

Q fa

ctor

(dB

)

Packet occupancy densitiy

off time ( s)

Q fa

ctor

dife

renc

e (d

B)

Figure 5.6: Q factor as function of the packet occupancy density. The lines are visual guides..

5.3 Raman amplification in hybrid WDM/TDM-PON

As said above, the use of combined multiplexing schemes, wavelength division

multiplexing/time division multiplexing (WDM/TDM) is pointed out as a possible

solution for the deployment for next generation access (NGA) networks. As an

example of NGA networks, the recently proposed scalable advanced ring-based

passive dense access network architecture (SARDANA) seeks to establish an optical

passive transparent infrastructure over a dense extended-range area, capable of

supporting future broadband services [1]. This kind of network transparently merges

TDM-tree sections connected to a metro WDM double-fibre ring, by means of passive

Remote Nodes (RNs), as depicted in Figure 5.7. This architecture finds a compromise

between WDM-PONs and TDM-PONs in terms of capacity and cost while offering

95


centralized management and bandwidth allocation from the OLT, simplifying the

TDM upstream protocol [9]. This is possible by using wavelength routing based

on two stages of matched arrayed-waveguide gratings (AWGs) that create a virtual

point-to-point connection between the OLT and the ONU. This network also provides

extended reach due to optical amplification, such as RFA and remotely pumped EDF.

The latter can be placed placed at the RNs or in line with the fiber and are pumped

from the CO while the optical fiber of the links provides Raman gain.

Figure 5.7: SARDANA architecture scheme.

Due to the use of a ring architecture, the network is able to provide traffic balance

through the shorter path and resiliency in case of fiber, splice, connector or component

failure, being the signals redirected for the other path. DS and US are wavelength

multiplexed, so each RN drops 2 DS signals and insert 2 US signals from/to the ring.

The number of RNs is a key parameter in terms of network performance because it

determines the number of wavelengths of the network and the total network capacity.

In terms of ring architecture and RNs, two scenarios are primarilly defined: rural

and urban. The rural scenario is conceived to operate over larger distances with a

smaller number of users. Typically, a maximal distance of 100 km can be achieved,

using 16 transmission channels, hence, a number of 256 ONUs can be reached.

For urban applications, the distances can be reduced but the number of users is

significantly increased. Thus the number of transmission channels duplicates, the

splitting ratio reduces to a half and the number of ONUs is then 1024. A list of

operational features for rural and urban scanerios is displayed in Table 5.2.

The principal aims of SARDANA network are primarily focused on deploying

C-band, mainly due to the availability of lower cost and mature devices in this

96


Table 5.2: Summary of possible scenarios of deployment of SARDANA network.

Scenario Ring (km) channels Splitter ONUs

Rural 80 16 1:16 256Urban 17 32 1:32 1024

spectral Band. Nevertheless, the extension of back-bone networks into the L-band

is driving a new development of active and passive devices adapted to operation in

this wavelength band. Those new devices are nowadays more expensive than their

related C-band devices, but they could be sold at competitive prices in a medium

term. Thus a bandwidth enlargement encompassing the C+L transmission band

is potentially very valuable due to the increase of the network capacity. Since the

latter is intended to have extended reach, the feasibility of amplification solutions

in C+L bands must be accounted for. Concerning the latter, this thesis was focused

mainly on the implementation in rural scenarios as summarized in Table 5.2, in both

operation modes. The simulation work presented bellow concerns the possibility

of obtaing enlarged gain in the WDM ring of the SARDANA network using solely

Raman amplification. This approach is analyzed in a normal and resilient operation

modes with a fixed pre-defined channel dropping order [29]. The OSNR is computed

and used as a performance indicator.

The first scenario (situation A) uses a bidirectional laser emitting at 1480 nm with

total power of 2 W (1 W in each direction for normal scenario) to pump 16 channels

spaced by 100 GHz (8 C band + 8 L band) comprised between 189.3 THz to 190.8 THz

in order to avoid the current GPON DS Video overlay (1550-1570 nm) as described in

Table 5.3. The channels leave the central office with an optical power of 0 dBm. The

transmission has accounted for that the bypass losses are 0.53 dB for channels and

1.01 dB for the pump. The simulated fiber is a SSMF with attenuations of 0.20 dB/km

and 0.25 dB/km for data and pump signals respectively.

Consider the general scheme depicted in Figure 5.7 with 8 RN. The normal scenario

is defined to be bidirectional, thus the data and pump signals are split in both

directions according to the following paths [RN1, RN2, RN3, RN4] and [RN8. RN7,

RN6, RN5].

The second analyzed scenario (situation B) also uses a bidirectional laser emitting

at 1480 nm from the CO to pump 16 channels, but unlike Situation A the total pumping

power was reduced to 1 W (500 mW for each side). In this situation, the channels

frequencies (8 C band + 8 L band) are also changed because the spacing was now

increased to 400 GHz comprised between 192.5 THz to 186.3 THz (see Table 5.4 ) and

the current GPON DS Video overlay was not avoided.

The results concerning situation A were only assessed using Raman amplification.

Thus, a previous simulation using the VPI simulator with all nonlinearities was

97


Table 5.3: Channel dropping order along the ring (2 channels per node) for Situation A.

RN number f1 (THz) f2 (THz)

1 190.8 190.72 190.6 191.53 190.4 190.34 190.2 190.15 189.4 189.36 189.6 189.57 189.8 189.78 190.0 189.9

Table 5.4: Channel dropping order along the ring (2 channels per node) for Situation B.

RN number f1 (THz) f2 (THz)

1 190.1 190.52 190.9 191.33 191.7 192.14 192.5 192.95 189.3 189.76 188.5 188.97 187.7 188.18 186.9 187.3

carried out, enabling the assumption that Raman scattering is the nonlinear dominant

effect and that spontaneous emission was the dominant source of noise. Thus, the

APA was implemented numerically to obtain the power of channels along the ring, as

well as the noise [30], [31]and [32]. These results are depicted in Figure 5.8.

In the resilient scenario, we consider that all the pump power and the 16 channels

travel in the same direction which can be counterclock wise [RN1, RN2, RN3, RN4,

RN5, RN6, RN7, RN8] or clock wise [RN8, RN7, RN6, RN5, RN4, RN3, RN2, RN1]. In

both paths the channels dropping order is the one given by Table 5.3. The results

concerning the net gain after dropping and OSNR are depicted in Figure 5.9 and

Figure 5.10.

On looking at the results, we notice that the L band channels are more amplified

than the C band which is expectable due to the maximal Raman gain coefficient by

pumping at 1480 nm. Thus a gain ripple of 3.2 dB is achieved in normal operation

scenario. We also observe that the OSNR is kept at very high value due to the

dominance of stimulated Raman scattering and that it decreases as the channels travel

over the ring. It must be noted that the pump depletion is enhanced in the propagation

in the first links where the channels are more amplified, as displayed in Figure 5.11

where the pump power at each node are displayed for the studied scenarios. On

98


1570 1572 1574 1576 1578 1580 1582 1584

6.0

6.5

7.0

7.5

8.0

8.5

9.0

9.5 Gain OSNR

Channel wavelength (nm)

Net

gai

n af

ter d

ropp

ing

(dB

)

57.6

57.7

57.8

57.9

58.0

58.1

58.2O

SN

R (dB

)

Figure 5.8: Net gain and OSNR spectra after dropping for normal scenario. Lines are guidesfor the eyes.

1570 1572 1574 1576 1578 1580 1582 1584

-6

-4

-2

0

2

4

6

8

10

12

14

16

Net

gai

n af

ter d

ropp

ing

(dB

)


Resilient counterclock wise Resilient clock wise

Figure 5.9: Net gain spectrum after dropping for resilient scenario. Lines are guides for theeyes.

looking the results regarding the resilient scenario, we observe that the gain varies

in a wider range and the nonlinear effects are expected to by very penalizing. So,

99


1570 1572 1574 1576 1578 1580 1582 158457.5

58.0

OS

NR

afte

r dro

ppin

g (d

B)

channel wavelength (nm)

Resilient counterclock wise Resilient clock wise

Figure 5.10: OSNR spectrum after dropping for the resilient scenarios. Lines are guides for theeyes.

0 1 2 3 4 5 6 7 8 9-30

-20

-10

0

10

20

30

40

Pum

p po

wer

(dB

m)

RN #

Normal Resilient counterclock wise Resilient clock wise

Figure 5.11: Pump power value in each node.

we can conclude that simple Raman amplification using a single 1480 nm pump

is not a solution for C+L transimission band in the context of WDM/TDM-PON,

such as SARDANA. An alternative solution could rely on hybrid amplifiers using an

100

5.4. Hybrid amplification schemes

association of distributed Raman amplification with in line remotely pumped EDFA.

The following section describes the potentialities of such amplifiers. An optimization

based on the dropping order and EDF span lengths was also accomplished in order to

enhance the obtaining of equalized gain.

5.4 Hybrid amplification schemes

The association of different kinds of amplifiers inside an optical fiber network

provides hybrid optimized solutions that make the most of each of their benefits,

namely wider optical bandwidth and and flat gain, using only a single 1480 nm

pump [33]. In metro-access network, such as SARDANA, the use of combinations

of hybrid amplifiers in the WDM ring can enhance the gain bandwidth enabling

possible solutions in the C+L spectral bands. In this thesis, we report some simulation

approaches that were carried out to enlarge the gain of SARDANA, but also to

optimize the gain of channels. However, the combined amplification scheme can

exhibit transmission issues in the WDM ring, such as power transients, due the to

add/drop. If EDFA are used in the RN, additional transients derived from the bursty

nature of upstream traffic can also occur and be very detrimental. Thus, the following

work is also addressed to those topics.

First, a sub-section introduces a brief theory about EDFAs, their numerical

implementation and an analytical modeling of their dynamic effects. We proceed

with experimental work that were carried on, namely the characterization of the

transients under different traffic features in terms of gain excursion and decay

times. Two mitigation schemes based on optical clamping are proposed and

implemented experimentally, being their feasibility for access networks analyzed. For

transmission at 10 Gb/s, the impact of hybrid amplifiers power transients on the

system performance was assessed experimentally. The possibility of extending the

transission band to C+L using hybrid amplifiers is also examined in a simulation

environment for 10 Gb/s transmission. An optimization strategy to equalize the gain

in the WDM ring is also proposed.

5.4.1 EDFA modeling

A charge distribution, such as Er3+, in a glass host originates a permanent electric

field. As a consequence, the Stark effect is induced, resulting into a splitting in the

energy levels [34]. Each of the energy levels has a total orbital momentum J that splits

into a manifold of energy sublevels g = J + 1/2. An energy diagram representing the

Stark effect in a three level laser is depicted in Figure 5.12. This splitting occurs with

uneven internal sub-population.

However, this system can be simplified into the three level energy system

101


Energy

N3

N2

N1

g3

l

g2

1

1

j

k

p1j

p2k

p3l

g1

ANR+ ANR-

Rlj Akj Wkj

Figure 5.12: Energy level diagram corresponding to a Stark split three level laser system. Thesymbols A±

NR indicate the thermalization between adjacent Stark sub-levels, while W and Adenote stimulated and spontaneous emission or absorption rates, respectively and R pumprate.

displayed in Figure 5.13, because the population distribution within the manifold

is maintained constant due to thermalization (Boltzmann distribution), enabling the

assumption of considering them as a single energy level [35].

N3

N2

N1

980 nm1480 nm 1520 - 1570 nm

=10 ms

1 s

Figure 5.13: Erbium ion energy-level scheme.

Due to uneven distribution within the manifolds, it is possible by pumping Er3+

glass to obtain an inversion of population between levels 1 and 2 and simplify this

system into a quasi-two-level system. This allows pumping at a wavelength near

102


1480 nm to provide gain in the 1525-1560-nm range [36]. This simplification will be

kept along this chapter.

When a signal ligth of intensity Is at wavelength λs traverses a slice of EDF of

infinitesimal thickness dz and atomic densities N1 and N2 for the lower and upper

energy levels respectively, the energy change dIs is given by:

dIs = σ21 (λs)N2 − σ12 (λs)N1 Isdz (5.2)

where σ12 (λs) and σ21 (λs) are the absortion and emission cross sections of the laser

transition at λs, respectively. The absorption and emission cross sections spectra for

alumino germanosilicate Er3+ glass fiber are displayed in the Figure 5.14. The data

were obtained from the VPI transmission maker librairies.

1400 1450 1500 1550 1600 16500.0

1.0x10-25

2.0x10-25

3.0x10-25

4.0x10-25

5.0x10-25

6.0x10-25

7.0x10-25

8.0x10-25

21,

12 (m

2 )

wavelength (nm)

emission absoption

Figure 5.14: Emission and absoption cross sections spectra for alumino germanosilicate Er3+

glass fibers.

Equation 5.2 can also be rewritten as dIs/dz = gIs, where g is defined as the

gain coefficient. Considering a total population density Nt, the relative inversion of

population is given by D = (N2 −N1)/Nt. Thus, g can be expressed as:

g =Nt

2σ21 (λs) (1 +D)− σ12 (λs) (1−D) (5.3)

When D = 1, all the ions are in the excited state N2 which means that a full

inversion of population is achieved. On the contrary, if all the ions remain in the

ground state, D = −1. Thus, D assumes values comprised between -1 and 1 which

103


corresponds to negative and positive values for the gain coefficient g. Figure 5.15

represents the gain coefficient spectra for differents values of D calculated using

the absorption and emission cross sections displayed in Figure 5.14 for Alumino-

germanosilicate Er3+ glass fibers. We noticed that D = −1 denotes that the medium

is absorving at all wavelengths because the gain coefficient is negative. However, as

the relative inversion increases, a positive gain coefficient is obtained for the longer

wavelengths.

1400 1450 1500 1550 1600 1650

-10

-5

0

5

10

gain

coe

ffici

ent (

m-1)

wavelength (nm)

D=+1

D=-1

Figure 5.15: Gain coefficient spectra for different relative inversion of population for analumino-germanosilicate Er fiber. D = −1 indicates that all the ions are in the gound state,while D = +1 denotes a fully inversion of population.

Considering that stimulated emission combined with spontaneous emission will

introduce changes in the upper level population, the model needs to account for the

variation of N2. Therefore, the spatial power evolution along the longitudinal axis for

an arbitrary number of k waves includes this effect. The following expression was also

corrected, accounting for the forward (+1) and backward (-1) propagating terms in uk

and the overlap factor Γk. This formulation was presented by Saleh et al [37] and is

from now on referred as the Saleh model. The equations 5.4 and 5.5 represent the rate

equation for the fractional population of the upper state N2 (z, t) and the change of

power in the kth beam, respectively.

∂N2 (z, t)

∂t= −

N2 (z, t)

τ−

1

NtS

N∑

1

uj∂Pj (z, t)

∂z(5.4)

where S stands for the active area and τ the spontaneous lifetime of the upper level.

104


∂Pk (z, t)

∂z= NtukΓk

[(

σ21k + σ12

k

)

N2 (z, t)− σ12k

]

Pk (z, t) (5.5)

Under steady state conditions, the implementation of the above equations can

follow the traditional numerical apporaches for ODE or alternatively use the average

power analysis method (APA) presented in chapter 3. We recall that this method

assumes that the powers are constant over an elemental section of fiber dz and

consequentely, an analytical solution is obtainable within it.

For EDFA modelling, the APA allows a recursive computation of the power at the

output of dz according to the following equations [38]:

P outk = P in

k Gk (z) (5.6)

Gk (z) = exp([(

σk12 + σk

21

)

N2 − σk12Nt

]

Γkdz)

(5.7)

Averaging 5.6 along the length of the elemental section, after having substituted

5.7, it is easily shown that for each section the average power associated with the kth

wavelength is:

〈Pk〉 = P ink

(Gk − 1)

ln (Gk)(5.8)

The calculation of N2 in equation 5.7 is given by:

N2 =Nt

∑Nk=1

(

〈Pk〉

P ISk

)

1 +∑N

k=1〈Pk〉

P ISk

(

σk21

σk12

+ 1) (5.9)

where the intrinsic saturation power , P ISk ; is equal to:

P ISk (λk) =

hνS

(σak + σe

k) Γkτ(5.10)

A simulation using the method described above was implemented to obtain the

net gain of 25 data signals comprised between 1520 nm and 1570 nm at the output of

an EDFA. The latter is forwardly pumped at 1480 nm with a power of 100 mW, being

the span of EDF 2.5 m, 5.0 m and 7.5 m. The simulation used an Erbium density of

7 × 1024 m−3, a overlap factor of 0.43 and a fluorescence lifetime equal to 10 ms. The

emission and absorption cross sections are the ones depicted in Figure 5.14, being the

spatial step , ∆z, equal to 2.5 cm. The gain results for the three lengths of EDF are are

plotted in Figure 5.16.

On looking at the results, we verify that longer EDF provide more gain in the

longer wavelength side. The results displayed in Figure 5.15 for the gain spetrum

105


1520 1530 1540 1550 1560 1570

-10

-5

0

5

10

Net

gai

n (d

B)

wavelength (nm)

L=2.5 m L=5.0 m L=7.5 m

Figure 5.16: Net gain spectra at the output of three EDFA with 2.5 m , 5.0 m and 7.5 m,respectively. The pump is a 1480 nm diode laser with 100 mW of power forwardly injectedinto the EDF.

demonstrate that an unpumped EDF has its higher attenuation values at wavelengths

around 1535 nm but the high emission cross sections in that spectral region allows

for higher gain when strong excitation is provided (higher inversion of population).

At longer wavelengths, a lower excitation level is required to have gain, but the

maximum gain is smaller. For longer EDF, the pump power decreases as it propagates

along the fiber and consequentely the ions in excited state decrease. Wavelengths that

require smaller inversion of population to provide gain are more amplified in that

context than they would be if smaller lengths of EDF were employed.

5.4.2 EDFA transients

The use of perturbation theory, allows the derivation of the EDFA temporal gain

dependency disturbed from the steady state. The approach explained bellow was

developed by Sun et al [39], [40] for the situation where the number of WDM channels

that are inputted in an EDFA vary as a consequence of add/drop. The same procedure

can be applied to a single channel with bursty/packeted data by assuming that the idle

times corresponde to a steady state with a number of channels equal to zero, disturbed

from the equilibrium with the addition of one channel.

We assume N WDM channels and a single stage EDFA with a length L. The powers

in this formulation are expressed in units of number of photons per unit time. The

106


fractional population in the upper state, N2 (z, t) and of the lower state, N1 (z, t), satisfy

the relationship N1 (z, t) +N2 (z, t) = 1.

The upper state population N2 can be found from equation 5.5 and substituted into

equation 5.4 according to:

τuk∂

∂t

(

1

Pk

∂Pk (z, t)

∂z

)

= −uk

(

1

Pk

∂Pk (z, t)

∂z

)

−1

P ISk

N∑

j=1

uj∂Pj (z, t)

∂z− αk (5.11)

where αk = NtΓkσ12k is the absoption constant. After integration over z from 0 to L and

by naming P ink (t) and P out

k (t) the input and output powers of the kth channel at time

t, the following expression is obtained [41]:

τd

dt

[

lnP outk (t)

P ink (t)

]

= lnP outk (t)

P ink (t)

−1

P ISk

N∑

j=1

[

P outj (t)− P in

j (t)]

− αkL (5.12)

By introducing the overall gain parameter, Gk (t) = ln [P outk (t) /P in

k (t)]; and the

total absorption constant, Ak = αkL, equation 5.12 can be rewritten as the following

ordinary differential equations:

P ISk =

[

τdGk (t)

dt+Gk (t) + Ak

]

= −

N∑

j=1

P inj (t) [exp (Gj (t))− 1] (5.13)

The system of ODE stated above can be reduced into a single equation by

combination of channels k and j:

τdGkj (t)

dt+Gkj (t) + Akj = 0 (5.14)

where Gkj = P ISk Gk (t)− P IS

j Gj (t) and Akj = P ISk Ak − P IS

j Aj .

The solution is trivial and given by Gkj (t) =(

G0kj + Akj

)

exp (−t/τ) − Akj , being

the initial conditions used to determine G0kj .

The gain in the jth channel is then expressed in terms of the gain in the kth channel,

as:

P ISj Gj (t) = P IS

k Gk (t)−(

G0kj + Akj

)

exp

(

−t

τ

)

+ Akj (5.15)

The substitution of equation 5.15 in equation 5.13 originates the following

expression:

107


τd

dtP ISk Gk (t) + P IS

k Ak =

−N∑

j=1

P inj (t)

exp[

P ISk Gk (t)

]

−(

G0kj + Akj

)

exp

(

−t

τ

)

+Akj

P ISj

− 1

(5.16)

Equation 5.16 is solved to determine the Gk (t). To determine the gain of the other

channels, the solution of 5.16 must be substituted in equation 5.15 using the condition

that in steady state EDFA at t = 0, we have G0kj + Akj = 0.

The overall gain parameter introduced above can also be expressed using the gain

and absorption per units of length as Gk (t) = gk (t)L. The parameters g0kj and αkj are

defined as:

g0kj = P ISk gk (0)− P IS

j gj (0) (5.17)

αkj = P ISk αk − P IS

j αj (5.18)

Thus, equation 5.16 takes the following form:

τd

dtP ISk gk (t)L+ P IS

k gk (t)L+ P ISk αk (t)L =

−N∑

j=1

P inj

exp

[

P ISk gk (t) + αkj

P ISj

L

]

− 1

(5.19)

Considering that the time dependent gain, gk (t), was disturbed from steady state

by a small dimensionless perturbation,gkt (t), according to:

gk (t) = gk0 [1 + gkt (t)] (5.20)

According to perturbation theory, the substitution of equation 5.20 in equation 5.19

originates two equations:

P ISk gk0L+ P IS

k αkL = −N∑

j=1

P inj [exp (gj0L)− 1] (5.21)

τd

dtgkt (t) + gk (t) = −gkt (t)

N∑

j=1

P inj

P ISj

exp (gj0L) (5.22)

A saturation factor, γ is defined as:

108


γ = −

N∑

j=1

P inj

P ISj

exp (gj0L) =N∑

j=1

P outj

P ISj

(5.23)

An averaged saturation factor can be used, given by:

γ =γ (0) + γ (∞)

2(5.24)

where γ (0) and γ (∞) are the saturation factors in the initial and final states.

By considering the saturation factor, equation 5.22 can be rewritten as:

τd

dtgkt (t) + (1 + γ) gkt (t) = 0 (5.25)

The solution is given by:

gkt (t) = g0kt (t) exp

(

−t

τe

)

(5.26)

where τe is defined by the following:

τe =τ

1 + γ(5.27)

Therefore, the gain per unit length can be expressed as:

gk (t) = gk0

[

1 + g0kt exp

(

−t

τe

)]

(5.28)

Equation 5.28 can be rewritten as equation 5.29 in order to include the initial and

final gains after the perturbation, enhancing its application range.

gk (t) = ~gk (∞) + [gk (0)− gk (∞)] exp

(

−t

τe

)

(5.29)

Where gk (0) and gk (∞) are the gain per unit length of the kth channel before and

after the transient, respectively and P outk is its output power.

To assess the impact of the use of EDFA using burst mode transmission, several

experiments were carried on. The latter aim to characterize the transient in terms of

gain excursion and time constants. A comparizon with distributed Raman amplifiers

was also accomplished [42].

The implemented experimental setup used for the transient assessment consists of

an external cavity laser peaking at 1549.32 nm and an optical power of 10.0 dBm,

followed by a polarization controller and a Mach-Zehnder optical modulator

operating at 2.5 Gb/s. The considered digital data are programmable packets with an

approximate duration of 26µs (pseudo random bit sequence (PRBS) of length 216 − 1)

and different idle times (4, 20, 40 and 80µs), produced by a BERT (Agilent N4901B).

109


After the Mach-Zehnder the optical signal is amplified by an EDFA, attenuated, and

then injected into a span of 40.0 km of SSMF. The optical power at the fiber input and

output are 10.0 dBm and 0.5 dBm, respectively. To analyze the temporal evolution

of the packets, a PIN (HP-11982A) and an (Agilent Infinium 86100A) oscilloscope

were used as receivers. The EDFA was previously characterized in order to determine

the input optical signal saturation power (-10 dBm for driving current of the pump

laser equal to 1.98 A for a threshold driving current of 0.7 A). It must be noted that

all the subsequent experiments were carried out with the EDFA in deep saturation

(corresponding to a pump power of 26 dBm). The gain excursion of packets was

analyzed using the oscilloscope.

Figure 5.17: 216 − 1 bit PRBS packet - 50000 bit idle obtained in the oscilloscope at the receiverfor a Raman amplification with 40 km of SSMF. The horizontal scale is 5.000 µs/div and thevertical scale is arbitrary.

For the Raman amplification assessment, the implemented experimental setup

uses the same scheme as the EDFA but an isolator was inserted after the Mach-

Zehnder modulator to protect the signal source. The EDFA and the attenuator were

removed and a Raman amplification module from Amonics, model ARA-CL-4B-R-FA,

with 4 pumping lasers, peaking at 1426, 1444, 1462 and 1487 nm with a total optical

power of 1 W was added to the setup. The experiment was carried out with a span of

SSMF fiber with 40 km. The input and output fiber powers are equal to -0.52 dBm and

3.22 dBm, respectively.

The results concerning the Raman amplification scheme are depicted in Figure 5.17,

110


with the visualized patterns at the oscilloscope. As excepted, the amplifier is

unsaturated so transients are not visible. However, when EDFA are used, a distinct

decaying on the packet envelope is assessible. The latter exhibit a behaviour that fit

an exponential decay, with characteristic time τch given by 5.30.

P (t) = P0 exp

(

−t

τch

)

+ P∞ (5.30)

where P , P0 and P∞ stands for the instantaneous optical power, the packet initial

optical power and the steady state optical power, respectively. Figure 5.18 (a), (b), (c)

and (d) display the observed packets and the exponential fitting curves for idle times

value equal to 2, 20, 40 and 80 µs, respectively.

The characteristic decay times obtained from the packet envelope fit (τch) are also

presented in Figure 5.19. Their dependency with the idle time was adjusted by an

exponential decay with r2 value equal to 0.9803.

From these results, we notice that the packet decaying time decreases with the idle

time increase, resulting from the increase of the Er3+ ions population inversion [43].

The exponential fit depicted in Figure 5.19 demonstrates that there is a stabilization

towards 1.35± 0.25µs as the idle time increases indefinitely.

The obtained experimental results were also compared to the theoretical model,

described above, namely the time parameter τe, given by 5.27. In the calculation, we

consider an EDF core radius equal to 1.8µm, a confinement factor of 0.9, being the

absorption and emission cross sections are respectively equal to 2.1 × 10−25m2 and

5.5 × 10−25m2 at the pump wavelengths and 5.2 × 10−25m2 and 5.5 × 10−25m2 at the

signal wavelength [44]. The packet initial and final optical power were used in 5.29

(gk (0) and gk (∞)). The analyzed scenario was the one with the highest idle time

(80µs) in order to obtain a good inversion of population and thus a situation close to

the steady state before the transient regime. The obtained value for the theoretical

characteristic decay time, τe, given by equation 5.29 was 1.72µs, which is in very good

agreement with the one obtained experimentally (1.71± 0.03 µs).

In the EDFA dynamics, the restoration of the initial conditions depends on the

idle time. If the latter is high enough to allow a complete inversion of population

of Erbium ions, the initial conditions are restored and the next packet will behave as

its precedent. However, if the next packet arrives before complete inversion, it will

experience less initial gain and consequently less power decay. To assess the amount

of gain excursion with variable idle time we need to evaluate how consecutive packets

are affected by variable idle time [45].

The experiment was extended in order to evaluate the sequences described in

Table 5.5. The patterns of the packets with 212 − 1 (Sequence A and Sequence B) and

216 − 1 (Sequence C and Sequence D) were visualized in the oscilloscope as displayed

in Figure 5.20 and Figure 5.21, being noticeable that the packets exhibit different gain

excursion due to the variable idle time. The initial power difference of consecutive

111


-2.0x10-5 -1.0x10-5 0.0 1.0x10-5 2.0x10-5 3.0x10-5 4.0x10-50.0

2.0x10-6

4.0x10-6

6.0x10-6

8.0x10-6

1.0x10-5

1.2x10-5

Po

we

r (W

)

time (s)

(a) Idle time equal to 2 µs.

-1.0x10-5 0.0 1.0x10-5 2.0x10-5 3.0x10-5 4.0x10-50.0

5.0x10-6

1.0x10-5

1.5x10-5

2.0x10-5

2.5x10-5

3.0x10-5

3.5x10-5

4.0x10-5

Po

we

r (W

)time (s)

(b) Idle time equal to 4 µs.

-2.0x10-5 -1.0x10-5 0.0 1.0x10-5 2.0x10-5 3.0x10-5 4.0x10-50.0

5.0x10-6

1.0x10-5

1.5x10-5

2.0x10-5

2.5x10-5

3.0x10-5

3.5x10-5

4.0x10-5

Po

we

r (W

)

time (s)

(c) Idle time equal to 40 µs.

-2.0x10-5 -1.0x10-5 0.0 1.0x10-5 2.0x10-5 3.0x10-5 4.0x10-50.0

5.0x10-6

1.0x10-5

1.5x10-5

2.0x10-5

2.5x10-5

3.0x10-5

3.5x10-5

4.0x10-5

Po

we

r (W

)

time (s)

(d) Idle time equal to 80 µs.

Figure 5.18: Visualized packets and their envelope exponential fit.

packets was then computed. For the lower length packets (212−1 PRBS) that difference

decreases with the idle time (21.81 µs for Sequence A and 3.76 µs for Sequence B). This

result was interpreted by considering that lower idle time situation is not sufficient to

obtain a good inversion of population of Erbium ions. The same tendency was also

observed for the longer length packets (Sequence C and Sequence D).

To assess the performance of a network in the presence of packet traffic, we have

analyzed the Q factor at the receiver and compare its value with the Q factor in

the back to back situation. The Q factor difference between the back-to-back and

the receiver (∆Q) was computed as a function of the packet occupancy density, as

depicted in Figure 5.22. The used sequences include packets with 216 − 1 intercalated

112


0 10 20 30 40 50 60 70 80 90

1,5

2,0

2,5

3,0

3,5

4,0

4,5

5,0

Pac

ket d

ecay

ing

time

(s)

idle time ( s)

packets with 216-1 PRBS @2.5 Gb/s exponential fit

Figure 5.19: Packet decaying time as a function of the idle time. The line represents anexponential fit in the form A0 exp (B0) + C0. A0 = 3.32 ± 0.27 µs, B0 = 36.39 ± 10.22 µs,C0 = 1.35± 0.25 µs, r2 = 0.9803.

Table 5.5: Tested bit sequences.

212 − 1 10000 212 − 1 50000Sequence A

PRBS zeros PRBS zeros

212 − 1 100000 212 − 1 200000Sequence B


216 − 1 10000 216 − 1 50000Sequence C


216 − 1 100000 216 − 1 200000Sequence D


with variable idle times in the 2-80 µs range (points 1, 3, 5 and 7 in Figure 5.22) and

packet with 216 − 1 intercalated with fixed idle times in the 2-80 µs range. By looking

at the plot, we notice that the best situations, the ones in which the ∆Q has a lower

value (points 6, 7 and 8), is obtained with a high packet density. This result is due

to the increase in the average power at the receiver. We also notice that the ∆Q is

quite stable in the 0.3 − 0.7 occupancy density range and that the gain excursion due

to variable idle time does not seem to be more penalizing than the situation with fixed

idle time.

113


Figure 5.20: Visualized sequences at the receiver: (a) Sequence C, (b) Sequence D. Horizontalscale 20.00 µs/div and vertical scale is arbitrary.

Figure 5.21: Visualized sequences at the receiver: (a) Sequence A, (b) Sequence B. Horizontalscale 5.000 µs/div and vertical scale is arbitrary.

5.4.3 EDFA transient mitigation

Mitigation methods for burst mode amplifiers transients were successfully

demonstrated and are based on a combination of techniques such as linear gain

control, gain clamping and fast automatic gain control [46], [39]. Additionally, passive

mitigation approaches have also been proposed, such as the use of EDF with a

large active area of Er3+ [47], [48]. In the optical domain, the idea of re-circulating

ASE as a clamping mechanism in a feed-back loop was presented in the context of

WDM ring networks [49]. The idea of re-circulating ASE using a bandpass filter

to select the lasing wavelength, and a variable optical attenuator (VOA) to adjust

the feedback cavity loss was presented in [50]. The same principle using FBGs as

bandpass filters was demonstrated to flatten and clamp the gain of L-band EDFA for

114


0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

Difference Back-to-back Transmission

Packet occupancy density

Q fa

ctor

2

3

4

5

6

Q factor

1

23 4

5

67

8

Figure 5.22: Q factor as a function of the occupancy density. The lines are visual guides.

WDM applications [51]. Other approaches based on feed-back techniques make use

of a DWDM multiplexer [52]. Published studies in the context of WDM ring PON,

such [51] have assessed the stabilizing effect of feedback based gain clamping on

cascaded EDFA. Hybrid electronic/optical techniques to achieve transient mitigation

are valuable such as the approach presented in [53] in which, the use of a packet

envelope technique to generate an inverted modulation on a control signal promotes

the mitigation of the transients. The subsequent use of a FBG filters the control signal.

In order to reduce the impairment of transients, we report two mitigations

techniques that were developed, based upon the optical clamping. The first method

performes clamping using the saturation feedback signal reflected on a FBG. Like

the previously reported feed-back methods, it provides constant gain on the packets

envelope. The second method uses bidirectional CW clamping as an enhanced

method.

Method 1

This solution preserves the passiveness of the access network, clamps the transient

of the packets gain and thus improves the performance of the network. The idea of

using a CW DFB laser signal in the feed-back loop instead of re-circulating ASE, as

fully used in the literature, is valuable because it increases the OSNR. This technique

is feasible in the context of hybrid WDM/TDM PON, namely in the TDM tree, by

using a small portion of the downstream to clamp the gain of the upstream bursty

115


channel. The proposed scheme is illustrated in Figure 5.23. It is composed of two

Figure 5.23: Scheme of the experimental setup used for the gain clamping.

distributed feed-back (DFB) lasers peaking at 1553.30 nm and 1550.88 nm, emitting

with optical power of 13.00 dBm and 3.54 dBm, respectively. The wavelength of the

control laser was previously tuned in order to match the Bragg wavelength of a FBG

with a rejection of 27.00 dB and a bandwidth, ∆λ, equal to 0.81 nm. The first laser

(data) is followed by a polarization controller, modulated by a Mach-Zehnder optical

modulator at 2.5 Gb/s, with programmable packets, as described in the previous

section, and coupled with the second CW laser. The two copropagating signals are

injected into an EDFA pumped with a laser diode at 965 nm. After the EDFA the

control signal is reflected by the FBG and re-injected into the EDFA generating a

lasing mechanism. The power of the feedback signal is controlled by a variable optical

attenuator (VOA) placed in the feedback path. The spectrum images captured by the

OSA at the points A and B assigned in Figure 5.23 are displayed in Figure 5.24. Note

that in point B, the clamping channel was removed by the FBG. The data signal is then

filtered, being its power adjusted by a second variable optical attenuator (VOA), in

order to have a constant value of -8 dBm at the receiver (C).

The experiment was accomplished with packets of 216 − 1 PRBS and idle lengths

equal to 50000 bits at 2.5 Gb/s and then repeated at 10.0 Gb/s where the idle length

was varied from 50 000 to 200 000 bits. The measurements include the packets gain

excursion and the Q factor. For comparison purposes, Q measurements when the

packets are replaced by PRBS with lengths equals to 215 − 1 and 231 − 1 were also

carried out.

For the measurements carried out at 2.5 Gb/s, oscilloscope captions were taken

at the receiver for increasing re-injected power (from -30 dBm to 5 dBm) and for the

back-to-back situation, as depicted in Figure 5.25. The packet gain excursion and the

Q at the receiver were measured being their dependency with the re-injected power

116


(a) Point A.

Removed by FBG

(b) Point B.

(c) Point C.

Figure 5.24: Spectra captured by the OSA. Horizontal scale 0.47 nm/div, vertical scale10 dB/div.

displayed in Figure 5.26. The back-to-back has a Q-factor equal to 5.85 dB. The Q-

factor obtained when packets are replaced by PRBS with lengths 231 − 1 and 215 −

117


(a) Back-to-back. (b) -30 dBm. (c) -25 dBm.

(d) -20 dBm. (e) -15 dBm. (f) -10 dBm.

(g) -5 dBm. (h) -2 dBm. (i) 5 dBm.

Figure 5.25: Spectra captured by the OSA. Horizontal scale 0.47 nm/div, vertical scale10 dB/div.

1 are also displayed. On analyzing those results, we notice that by increasing the

power of the feed-back signal, the clamping is enhanced and consequently, the gain

excursion of the packets decreases. From the Q-factor results, we also observe that the

clamping improves the performance of the system. This effect observed in packets is

not observed in PRBS where the Q is almost constant.

For the measurements carried out at 10 Gb/s, a similar behaviour is also observed:

the gain excursion decreases with the feedback signal power and consequently, the

system performance is also enhanced. Measurements with different idle lengths

(corresponding to 5, 10 and 20 µs) were also accomplished. The results concerning

the gain excursion as a function of the measured reinjected power are displayed in

Figure 5.27. For longer idle times (20 µs, 200000 bits), the variation of gain excursion,

when the feedback signal power is varied from -30 dBm to -10 dBm, is higher and the

packet still exhibit transient when the packets with lower idle times (5 µs and 10 µs)

118


-35 -30 -25 -20 -15 -10 -5 0 5 10

0

1

2

3

4

5

6

7

8

PRBS 231-1 PRBS 215-1 Packet

Re-injected optical power (dBm)

Pac

ket g

ain

excu

rsio

n (d

B)

2.5

3.0

3.5

4.0

4.5

5.0

5.5Q

-factor (dB)

Figure 5.26: (left) Packets gain excursion as a function of the re-injected signal power. (right)System signal-to-noise ratio as a function of the re-injected signal power with traffic based onpackets and PRBS of length 231 − 1 and 215 − 1. The receiver power is kept constant equal to-8 dBm.

are already fully clamped. This result is explainable by considering that the increase

in the transient is due to an increase in the Er3+ ions population inversion. The system

performance is assessed with the Q-factor results depicted in Figure 5.28. The results

are consistent with the gain excursion, showing that the Q-factor is improved when

the gain excursion is decreased.

The transients in EDFA arise due to the modulation in time, namely the packets. As

the packets bit rate approaches the frequency with which the erbium gain dynamics

can respond, the transients become more relevant. In both situations (2.5 Gb/s and

10 Gb/s), the packets have the same lengths (215−1) but they have different durations

(26.21 µs and 6.55 µs) which corresponds to different dynamics with the Erbium. The

input with the longer duration will quickly deplete the reservoir by causing a very fast

power transient in a way that is more noticeable to the packet with shorter duration.

The idle time is also important because it is concerned with the recovery time of the

Erbium dynamics. The longer the idle time the higher is the inversion of population

of Erbium ions when the next packet arrives and the stronger the transient. Thus the

situation that we can compare is the one with the same idle time (20 µs). The results

at 2.5 Gb/s in Figure 5.26 were obtained with idle time equal to 20 µs while the results

at 10 Gb/s (Figure 5.27 and Figure 5.28) were obtained with idle times equal to 5, 10

and 20 µs. To assess the influence of the packet duration on the EDFA transients, only

119


-30 -25 -20 -15 -10-1

0

1

2

3

4

5

6

7

8

Pac

ket g

ain

Exc

ursi

on (d

B)


idle time 5 s 10 s 20 s

Figure 5.27: Gain excursion as a function of the feedback signal power for packets with idletime equal to 5 µs (50000 bits), 10 µs (100000 bits) and 20 µs (200000 bits) at 10 Gb/s. Thereceiver power is kept constant equal to -8 dBm.

the 20 µs idle time situations can be compared. Hence it is expectable that the longer

packets present stronger transients measured by higher gain excursion and lower Q

factor. On looking at Figure 5.18, we verify that the characteristic decay times for idle

time equal to 20 µs is approximately 3.5 µs. Since each of the packets (at 2.5 Gb/s and

10 Gb/s) have a duration superior to that time, we can conclude that both the packets

are affected in the same way by the transients. The results on Figure 5.26 and Figure

5.27 demonstrate a gain excursion of 6.51 dB and 6.54 dB respectively, being the packet

at 10 Gb/s mitigated with lower re-injected power. On looking at Q factor, we notice

that it the same trend is observed at both bitrates. At 10 Gb/s the Q-factor stabilizes

with lower re-injected power.

Method 2

The idea of this technique is to use the intrinsic characteristics of bidirectional

traffic existing in PON as a transient mitigation mechanism for the packeted /

bursty upstream traffic. In such architecture, instead of using re-circulating ASE that

degrades the OSNR, the downstream channel itself acts as a clamping signal. Since

we presume that its power is possibly low due to the PON operation and that by this

reason the clamping on the EDFA transient could be insufficient, a FBG was inserted

in order to enhance the clamping. Unlike conventional clamping mechanisms, the

proposed method does not use feed-back. The principle of operation is summarized

120


-30 -25 -20 -15 -101.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

Q fa

ctor

(dB

)


idle time 5 s 10 s 20 s

Figure 5.28: Signal-to-noise ratio as a function of the feedback signal power for packets withidle time equal to 5 µs (50000 bits), 10 µs (100000 bits) and 20 µs (200000 bits) at 10 Gb/s. Thereceiver power is kept constant equal to -8 dBm.

in Figure 5.29, where a remote node is schematized. In such a configuration, two

channels are dropped (and added) using the 200 GHz and 100 GHz add/drops. The

90/10 power splitters are used to select the downstream channel power so that it can

be used to clamp the EDFA. A FBG tuned with the clamping channel is inserted in the

opposite side of the EDFA to provide the bidirectional clamping.

Although the scheme suggests backward clamping for a matter of simplicity,

forward clamping is also feasible. To infer the efficiency of forward and backward

clamping, some simulations were carried on. They compute the packet gain excursion

by implementation of Bononi et al dynamical model for EDFA [43]. The results

obtained for a span of 15 m of EDF using 20 µs packets are plotted on Figure 5.30.

On looking at the results, it is noticeable that despite the forward clamping is

more efficient on mitigating the transient, the method converges when the clamping

power is increased. To demonstrate experimentally the proposed mitigation approach,

we implemented the setup depicted in Figure 5.31 that provides forward clamping.

Although, some of the used equipments are not cost efficient for PON, the experiments

serves the purpose to demonstrate the technique in a laboratorial environment.

It consists of two DFB lasers peaking at 1554.92 nm and 1550.80 nm with optical

powers of 13.0 dBm and 3 dBm, respectively, followed by polarizations controllers.

The laser at 1554.92 nm works as the upstream data signal and is optically modulated

by a Mach-Zehnder operating at 2.5 Gb/s with an extinction ratio of 14.4 dB. The data

121


Add/Drop200GHz

Add/Drop100GHz

Î»2Add/Drop 100 GHz

Add/Drop200GHz

US

DS

ch1ch2

ch1 ch2

ch2

90/10 10/90

ch1

Bragg wavelength=ch1

EDFA EDFA

Bragg wavelength=ch2

Figure 5.29: Generic clamping scheme feasible in a WDM/TDM-PON remote node (RN) inwhich two channels are dropped / added.

are programmable packets with a duration of 26 µs (pseudo random bit sequence-

PRBS of length 216 − 1) and an idle time equal to 20 µs (50000 bits), produced by

a bit error rate tester (BERT) (Agilent N4901B). The signal at 1550.80 nm works in

continuous wave (CW) and represents the clamping control signal, being its power

adjusted by a VOA. The two signals are brought together by a 50 / 50 coupler and

injected into a bidirectionally pumped highly EDF (Thorlabs ER-4/125) with a span

of 22.5 m. The average inputted power of the data signal is -7.0 dBm. The EDF

pumps are three diode lasers peaking at 1470 nm, 1480 nm and 1490 nnm with a

total injected power of 24.0 dBm. Two WDM couplers are used to join signals and

122


-10 -8 -6 -4 -2 0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Gai

n ex

curs

ion

(dB

)

CW power (dBm)

L=15 m forward clamping backward clamping

Figure 5.30: Packet gain excursion as function of control signal power for an EDF with 15 m.(black squares) fForward clamping, (red circles) backward clamping.

1554.92 nm

22.5 m EDF

50/50 WDM

Pump1480 nm

Pump 1490 nm

1550.88 nm

Pump1470 nm

FBG

VOA

DFB

DFB

WDM

50/50

PG2.5 GHz

Figure 5.31: Scheme of the implemented experimental setup.

pumps as well as three isolators to protect the sources. After the amplification stage,

a FBG with a rejection of 27.0 dB and an optical bandwidth, ∆λ, equal to 0.81 nm

is inserted in order to reflect the clamping control signal backwardly into the EDF. To

analyze the temporal evolution of the upstream data packets, a PIN (HP-11982A) with

123


responsivity of 0.7 A/W and an oscilloscope (Agilent Infinium 86100A) oscilloscope

were used. The gain excursions of packets as well as the Q factor were experimentally

analyzed.

Figure 5.32: Back-to-back. (Left)-Pattern- horizontal scale 10.00 µs/div, vertical scale 30mV/div. (Right) eye diagram- horizontal scale 200 ps/div, vertical scale 30 mV/div.

Figure 5.33: Receiver output with the CW signal off. (Left)-Pattern- horizontal scale 10.00µs/div, vertical scale 100 mV/div. (Right) eye diagram- horizontal scale 100 ps/div, verticalscale 100 mV/div.

Firstly, the experiment began with the visualization of the back-to-back

performance. Thus, with the CW signal off, the patterns and eye diagrams were

visualized on the oscilloscope, being the extinction ratio and Q-factor measured. After

that, the same procedure was applied at the system receiver. The measured Q factor

are 4.83 dB and 2.63 dB for back-to-back and receiver respectively. The captured

patterns and eye diagram on back-to-back and receiver are depicted in Figure 5.32

and Figure 5.33, respectively.

In order to emphasize the effect of the FBG in the proposed mitigation setup, we

compare the results obtained with FBG (enhanced clamping) and without FBG (simple

clamping) the FBG and varying the clamping control signal power -25.74 dBm to -

1.67 dBm, being the Q-factor measured.

The use of a control signal is able to clamp the gain whenever its power is high

enough, as stated from the square plot on Figure 5.34, where the gain excursion

124


is plotted as a function of the clamping control input power for the implemented

situations: simple clamping and enhanced clamping. Hence, a decrease on the gain

excursion is observed when the control power is increased. However, the use of a

FBG with Bragg wavelength coincident with the control signal is able to re-inject it

backwardly into the fiber, enhancing the clamping while keeping the same amount

of power. A decrease on the gain excursion is observed, when the control power is

increased. Both results were linearly fitted. For the simple clamping, the linear fit

provides a slope equal to -0.047 dB/dBm, the enhanced clamping data has a slope

equal to -0.036 dB/dBm. In the latter case, the gain excursion decreases 1.12 dB, when

the control power is equal to -25.7 dBm.

-30 -25 -20 -15 -10 -5 00.0

0.4

0.8

1.2

1.6

2.0

2.4

simple clamping enhanced clamping

Gai

n ex

curs

ion

(dB

)

Clamping control signal power (dBm)

1.12 dB

No mitigation level

Figure 5.34: Packet gain excursion as function of control signal power. Continuous lines arelinear fits. (squares) - Simple clamping, adjusted r square= 0.79; (circles) enhanced clamping,adjusted r square = 0.73. The dashed line stands for the gain excursion level for the unclampedsituation.

The clamping effect is also visible in the Q-factor that exhibits an improvement

with the control signal power. These results are displayed in Figure 5.35, where the

Q-factor as function of the control signal power is plotted for simple clamping and

enhanced clamping.

On looking at the results, we verify that the simple clamping shows two distinct

regimes. The first one, observable for low powers (until -15.7 dBm), demonstrates

that the Q-factor remains stable and the clamping is not assessable. However, from

that power value the Q-factor starts to grow linearly with the clamping control

signal power. The interpretation of this behavior is based on the idea that, bellow

125


-30 -25 -20 -15 -10 -5 03.0

3.2

3.4

3.6

3.8

4.0

4.2

4.4

4.6

4.8

Regime 2

w/o FBG with FBG

Q-fa

ctor

(dB

)

Clamping control signal power (dBm)

Regime 1

Figure 5.35: Q factor at the receiver as a function of the clamping control signal power.(squares) - Simple clamping and (circles) enhanced clamping. The dashed line stands for theQ-factor level for unclamped situation.

a predefined value of power, the effect of the clamping signal is negligible. Above this

value, the clamping has an effect on decreasing the gain excursion and the Q-factor

increases. The same behavior is not observable for the enhanced clamping in which

the Q factor remains almost constant with an average value of 4.71 dB in the range of

the inputted control powers. By decreasing the gain excursion over a threshold value

the Q-factor was definitely increased. It must be noted that, the use of the FBG was

able to improve the Q-factor to a value close to that of back-to-back values.

The dynamics in terms of gain excursion are determined by the packet duration

and idle times. By increasing idle times, the EDFA restores the initial conditions more

effectively and thus the gain excursion on the next packet envelope that income the

EDF is larger. By increasing the packet duration, the depletion effect on the excited

Erbium ions population is also more significant. If the system is not given enough

time to restore the initial conditions, the next incoming packet will experience less

gain and consequently less gain excursion. So, the results in terms of gain excursion

are a combination of these factors. However, the exponential dependency of the filling

and depletion effects of the Erbium excited state population impose a stabilization

effect [43] so asymptotic behaviors can be settled.

Hence, although, the method was analyzed by considering a specific situation

of packet and idle times, it is expected that the results complying other traffic

characteristics will only differ in terms of the necessary clamping powers. Thus, the

126


mitigation technique can be seemingly extrapolated to different types of packet traffic.

5.4.4 Transmission at 10 Gb/s

In this work, we assess the dynamic effect of adding/dropping channels in a

30 km, 10×10 Gb/s WDM system using a hybrid Raman/EDFA amplification scheme.

To reproduce the addition and dropping, the channels power function is given by a

square wave with low frequency. The hybrid amplification scheme is composed by a

multi-backward pumped Raman amplifier and an EDFA acting as preamplifier [54].

Figure 5.36: Scheme of the implemented experimental setup.

The experiment follows the setup depicted in Figure 5.36. Channel X (test channel)

is provided by a tuneable laser (Net Test Tunics) centred at 1546.71 nm with an input

power of 5 dBm, whereas the other DWDM channels are provided by nine DFB lasers

(Anritsu MT 9812B) comprised between 1547.72 nm and 1554.15 nm (100 GHz spaced)

and with an input power equal to 9.3 dBm. Each channel is followed by a polarization

controller with attenuator. The test channel is modulated at 10 Gb/s, while the nine

channels are modulated at 250 Hz by two Mach-Zehnder external modulators (Fujitsu

FTM+174M-5208-J155). The imposed modulation is a pseudo random bit (PRBS)

sequence of length 233−1, provided by a pattern generator (Anritsu MP1761B). The 10

channels are then coupled and injected into the 30 km of SSMF fiber. The polarizations

of the channels as well as the bias voltage of the modulators were adjusted to keep

all the channels approximated with the same optical power and signal to noise ratio,

measured before the coupler. An isolator was placed after the coupler to protect the

signal lasers from the backward propagated high-power Raman pumps. The Raman

127


pump module (Fitel HPU-CL12W14D-00077) was placed after the transmission fiber

and has a total of 14 pump diode lasers, [1410.0, 1410.2, 1416.9, 1417.3, 1452.1, 1458.1,

1465.5, 1472.6, 1494.8, 1502.4 nm] with a maximum pumping power of 1.175 W. The

power of each laser was tuned individually to keep the spectral gain region equalized.

To prevent the effect of chromatic dispersion on channel X a negative dispersion

fiber module (11268 m) was used before the receiver, this fiber length compensate

completely the dispersion of the SSMF link. In this scenario, the fiber attenuation

is controlled by the Raman amplifier module and the chromatic dispersion is

compensated by the additional fiber. The receiver is composed by an EDFA, a

filter centred in the channel X, a photodetector and an error detector (Anritsu

MP1762A). The signal electric and optical signals at the receiver were analyzed using

an oscilloscope (Agilent Infinium 86100A) and an Optical Spectrum Analyzer (OSA

Advantest 8384), respectively. After the optical filter a tunable attenuator (VOA)

was inserted, to change the optical power injected into the receiver. The considered

acquisition time for the BER measurement was 30 s, resulting in a BER floor of 10−13.

Each BER measurement was repeated three times and an average value was estimated.

Initially, we identify an error free regime (BER <10−12) and then successively decrease

the output power, using the attenuator, to obtain lower BER values. To appraise

Figure 5.37: Optical spectrum at the output of (a) the negative dispersion fiber and (b) thereceiver. The vertical and horizontal scales are 10 dB/div and 2.00 nm/div, respectively.

the impact of the addition and the dropping of channels, we set initially all the nine

channels on and then turn them off with the test channel always kept active. In the off

situation, only the test channel is amplified and consequently the transience because

of the other channels is not present. With the nine channels, the effect of transience

on the test channel is observed since the low frequency modulation reproduce the

behavior of the addition and dropping of channels in the system. The back-to-back

128


measurement was also accomplished.

The optical spectra obtained at the DCF fiber output, and the photodiode input

are displayed in Figure 5.37. These images show that the 10 DWDM channels are

equalized in terms of optical power. However, in the receiver is observed the channel

X and also a secondary peak not completely eliminated by the optical filter but with a

25-dB suppression.

Figure 5.38: Eye diagram at the receiver when all the low frequency modulated channels are(left) off and (rigth) on.

0.0040 0.0045 0.0050 0.0055 0.0060 0.0065 0.00700.0

5.0x10-5

1.0x10-4

1.5x10-4

2.0x10-4

2.5x10-4

3.0x10-4

3.5x10-4

Am

plitu

de (V

)

time (s)

Figure 5.39: Signal pattern, of the test signal, in the presence of add/drop function. The verticalscales are arbitrary.

129


The eye diagrams at the receiver output obtained with the low frequency

modulated channels off and on are depicted in Figure 5.38. The test signal pattern

in the presence of the low frequency modulated channels is also depicted in Figure

5.39. The oscilooscope image file was edited and smoothed using the Savitzky-

Golay method. The transient effect is clearly visible. The results concerning the BER

measurements as a function of the received power are shown in Figure 5.40. From

these results, it is noticeable that the power penalty for a BER of 10−9 is 8 dB. It is also

noticeable that the back to back curve and the channels off curve are coincident. In

the latter situation, we can conclude that the transmission channel do not introduce

any power penalty at this bitrate. However, with the add/drop of channels, the

observed penalties are due to the inexistence of power transients. Those transients

will enhance the degradation imposed by nonlinearities and amplified spontaneous

emission. In the worst case scenario with 9 drop-add channels, the power increase

due to channel dropping, degrades the performance of surviving channel because

of self-phase modulation in single mode fiber. The channel re-adding will result in

the degradation of all the channels because of cross-phase modulation and four-wave

mixing, which is enhanced when the chromatic dispersion is compensated.

-22 -20 -18 -16 -14 -12 -101E-12

1E-11

1E-10

1E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

back to back chanells on channels off

BE

R

Power at receiver (dBm)

Figure 5.40: BER versus the optical power at the receiver. The squares represent the back-to-back, circles represent all the channels on and triangle shown the situation where onlychannels X is on. Lines are guides for the eyes.

A second experiment was carried on at 10 Gb/s in order to assess the impact of

single channel Raman transients on the system performance [55].

The signal emitted by the tuneable laser (NetTest Tunics) is centred at 1546.71 nm

with input power equal to 10 dBm. The receiver is composed by a packet receiver,

130


an error detector (Anritsu MP1704A), a demultiplexer (Anritsu MP1802A) and an

oscilloscope (Agilent Infinium 86100A). The bit data are generated at 9.95328 GHz.

A packet with a 192 bits header (Preamble: 64, Frame: 64 and Sequence field size:

64), a 8840 bits payload and a gap length with 524288 bits (that corresponds to

approximately to 53 µs) is generated. The transmission medium is a SSMF fiber - two

different spans were used with 29425 m (fiber1) and 28816 m (fiber 2) respectively.

The Raman module (Fitel HPU-CL12W14D-00077) was used with the 14 lasers on

maximum power (1.2 W) and 11268 m a negative dispersion fiber was included to

prevent the penalties dues to chromatic dispersion. The patterns of the packet and eye

diagram visualized on the oscilloscope at the end of transmitter and the receiver for

each fibre are depicted in Figure 5.41.

Figure 5.41: Pattern and eye diagram at the end of the: (a) Tx, (b) Rx with fiber 1 and (c) Rxwith fiber 2.

To obtain the BER curves as a function of the power at the receiver an attenuator

(Agilent 8156A) was added to the experimental scheme. The power penalties obtained

for both fibers are approximately equal to 2.42 dB, as displayed in Figure 5.42.

The results obtained from this experiment infer that contrarily to EDFA that

were characterized, Raman amplification did not present any power transients.

131


-33.0 -32.5 -32.0 -31.5 -31.0 -30.5 -30.01E-12

1E-11

1E-10

1E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

0.01

back to back fiber 1 fiber 2

BE

R

Power at receiver (dBm)

2.42 dB

Figure 5.42: BER curves for back to back, fiber 1 and fiber as a function of the power at receiver.

However, both theory and experiments have demonstrated that a single channel

Raman amplifier can exhibit transients under saturation conditions [22]. Therefore,

we can conclude that the tested Raman amplifier was still unsaturated, despite the

high amount of injected pumping power.

We will now return to the WDM ring part the SARDANA network and analyze

hybrid Raman/EDFA amplification schemes for transmission at 10 Gb/s. The scenario

studied is simular to the one described in section 5.3. WDM ring 80 km long with

8 equally spaced nodes conceived for rural scenario in normal operation mode. In

the central office (CO), a bidirectional laser emitting at 1480 nm is used to pump 16

channels spaced by 400 GHz (8 C band + 8 L band) comprised between 192.5 THz to

186.3 THz. The channels leave the central office with 0 dBm of optical power. The

transmission has accounted for that the bypass losses are 0.53 dB for channels and

1.01 dB for the pump, being the attenuation of channels and pump are 0.20 dB/km and

0.25 dB/km, respectively. The channels drop losses are 3 dB. In the above described

frame, two situations are analyzed, one with Raman amplification and the other with

a hybrid amplification scheme composed by in line EDF with Raman. Hence, an

optimized span of Erbium Doped Fiber (EDF) is inserted in the mid length of each link

according to the dropping channels powers. This procedure was performed with total

pumping power equal to 1 W (500 mW in each direction). The results are compared in

terms of gain ripple and total amount of EDF span, being their suitability for practical

PON analyzed [56].

The 16 analyzed channels with the exception of the last two rely on the maximal

132


bandwidth of Raman gain efficiency; see Figure 5.43 where the Raman gain efficiency

spectrum is plotted for a SSMF pumped at 1480 nm, being the C and L channels

represented by black and red arrows, respectively.

1500 1520 1540 1560 1580 1600 1620 16400.0

1.0x10-4

2.0x10-4

3.0x10-4

4.0x10-4

5.0x10-4

6.0x10-4

7.0x10-4

8.0x10-4

9.0x10-4

Ram

an g

ain

effic

ienc

y (W

-1m

-1)

Channel wavelengths (nm)

Figure 5.43: Raman gain efficiency for pumping at 1480 nm. The channels are representedby arrows (black-C band and red-L band). The curve was obtained by interpolation ofexperimental data [57].

The strategy followed for the optimization is the following: (i) drop the channels

with the higher gain firstly to settle a maximal gain level and decrease the effect of

pump depletion, (ii) whenever the channel power reduction surpasses a predefined

value, try several span of EDF fiber in the mid-link to minimize the gain ripple.

This method is simple because it is based on the optimization of only one variable,

the EDF length. The Raman amplification is maximized in the first links by the

settings on the dropping order.The obtained optimized results were also compared

with simple Raman amplification. The simulation is based on the implementation of

Raman propagation equations [33] and the Saleh [37] model for EDFA using the semi-

analytical average power analysis method [58] [38]. This method enables us to obtain

quick and accurate solutions.

The optimal dropping order is listed in table 5.4, which corresponds by dropping

firstly the channels in the maximal Raman gain efficiency and then move outwardly,

as assigned in Figure 5.43. The results for total pumping at 1 W (500 mW for

each direction) are depicted in Figure 5.44. The top graph displays the power after

dropping spectra for optimized hybrid Raman/in line EDFA and simple Raman, being

the middle graph the optimized EDF span and their position in the ring (in relation

133


-9-6-303

1550 1560 1570 1580 1590 1600 1610

Net

gai

n (d

Bm

)


0369

1215

ED

F sp

an (m

)

-40 -30 -20 -10 0 10 20 30 4005

101520

Pum

p po

wer

(dB

m)

Distance to CO (km)

Figure 5.44: (Top)- Net gain dropping spectrum after dropping. Squares- simple Raman.Circles- optimized hybrid Raman/ in line EDFA. The horizontal dashed line settles a detectionthreshold. (Middle)- Optimized spans of EDF and their positions along the ring. The verticaldashed lines are used to place the dropped channel wavelength along the ring. The total pumppower is equal to 1 W. (Bottom) - Available power to pump the EDF along the ring.

to the central office CO). The bottom graph displays the power available to pump the

spans of EDF. On looking at the simple Raman results, we verify that just by choosing

a dropping order compliant with the maximal Raman gain efficiency, the power is

considerably flat in the [1567-1590] nm range and then decreases as the channels

wavelength move away from the maximal gain efficiency. On those aparted links, the

insertion of span of EDF fiber is able to increase to power above the 0 dBm threshold.

It must be noted that in order to provide gain in L band higher spans of EDF are

used. On looking at the pump power results, we notice that due this methodology,

the available pump power in distant links is still high enough to pump the EDF spans.

Hence, in the hybrid approach a total span of EDF equal to 22 m is used (2 m for C

band amplification and 20 m for L band amplification) to attain a ripple of 2.54 dB

over a bandwidth of 50 nm.

134

5.5. Chapter summary

5.5 Chapter summary

This chapter described the work that was carried out concerning applications of

optical amplifiers in access networks. It starts with an overview of passive optical

networks, namely their principles of operation and transmission techniques (TDM-

PON, WDM-PON). The topic of extended reach PONs is also addressed, presenting

state of the art achievements and research challenges. Hybrid WDM/TDM PONs,

such as SARDANA used to metro and access are then presented and discussed. We

remark that most of the work developed in the frame of this chapter was planned for

this kind of architecture. Although SARDANA was initially planned for operation

in the C band, the possibilibity of extending the transmission to L band in C+L

applications is discussed and amplification issues are analyzed. By implementing

simple Raman amplification for the WDM ring part of the SARDANA network, it

was concluded that this amplification scheme was not suitable for C+L transmission.

Thus, hybrid Raman/EDFA amplification was considered.

The second part of this chapter is about hybrid amplifiers. An introduction

to EDFA theory is presented. Their numerical implementation and dynamical

behaviour is also approached. Since the upstream traffic in SARDANA network

is bursty/packeted, issues related to the used of amplifiers, Raman, EDFA and

both are studied. Experimental work was carried on to evaluate their impact,

and, for EDFA, dynamic behaviour was inferred from the theory. Besides that,

mitigation techniques based on optical clamping were presented and implemented

experimentally. The chapter ends with an analysis on 10 Gb/s transmission systems,

where the impact of combined Raman/EDFA transients are assessed on system the

performance. Hybrid amplification is also addressed for the WDM ring of SARDANA

and an optimization strategy is presented to optimize the channel gain in an extended

C+L band transmission scenarios.

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141

Chapter 6

Conclusions

6.1 Conclusions

This thesis has provided a survey of applications of Raman fiber amplifiers in the

context of modern communications. Two general topics were addressed:

1. the development of an efficient simulator for Raman amplified links and

an optimization algorithm for multipump allocation to obtain enlarged and

equalize gain in WDM transmission systems.

2. the use of Raman amplification alone and in hybrid combinations with EDFA in

metro and access networks and decurrent dynamic effects.

Concerning the first topic, several issues were surpassed, namely the difficulties

associated to mathematical description of Raman amplifiers, a boundary value

problem with non-linear ordinary differential equations. Although, several numerical

approaches are already known in the litterature to solve this kind of problems, Raman

equations present some drawbacks that disable the use of traditional solvers. It

was mentioned earlier that simple shooting methods fail due to the instability of the

Raman equations with initial conditions. Therefore, the work developed in this thesis

began with a study of numerical methods for non-linear ordinary differential equation

with boundary value conditions. It was demonstrated that modified shooting and

collocation method were suitable approaches but a simulator could benefit from the

use of semi-analytical method in terms of efficiency. Thus, the average power analysis

was implemented together with an optimization routine for backward or bidirectional

pumping architecture. Our mathematical study proceed with the a stability analysis of

critical points in order to determine the asymptotic behabiour of Raman equations. It

was shown that the only critical point with physical significance is the origin and that

this point is a node asymtotically stable. The same conclusion was also obained using

Lyapunov second method. A geometrical interpretation of this result is assessible

from the representation of solutions trajectories on the phase portrait.

One objective of this thesis is to provide broadband and equalized solutions for the

data signals gains. It is known from Raman theory that the use of multipump schemes

in both forward and/or backward configurations is able to accomplish this goal as

6. Conclusions

long as the pumping parameters (powers and wavelengths) are adjusted. Hence,

an optimization algorithm to accomplish this objective is imposed. There are several

dificulties arising from this problem, namely the ones derived from the multivarible

optimization. Aware of this issue, our first approach was to use an heuristic and

genetic algorithm was a good candidate because of its robustness and ability to obtain

solutions on multivariable, multiobjective problems with constraints. The issues

of this method is the required computational effort to obtain solutions, so for the

purpose of our problem they can be obtained in a timescale superior to the one needed

for networks reconfiguration. To surpass this impediment, we first dimensioned

the genetic algorithm in terms of operators (selection, crossover and mutation) and

population size to increase its efficiency in solving our problem. Selection itself

is a very important operator because it decides how fast the solutions converge to

a solution. Stochastic uniform, that we tried to prove to be the more the more

efficient selection method for our problem , mimics the statistical results expected

for population with a high number of individuals, without expense of efficiency.

Dispite our efforts in dimensionsing GA, we verify that a hybrid combination with

a local search method, such as Nelder-Mead method could increase significantly the

efficiency while keeping the accuracy of the solutions. Thus, using the two methods

together, the advantages of both are combined: the convergence of GA and the high

accuracy of Nelder-Mead. Dispite that, it was also demonstrated that by chosing

properly the rigth moment to switch from one method to the other could also increase

the effeciency of the algorithm. For the particular problem of a 5 backward pumped

Raman amplifier with 16 data signals speach over a bandwidth of 50 nm, the GA was

switched to Nelder-Mead after 17 generations were evolved. This coresponds to a

solution obtained in half the time that would be necessary if the hybrid method were

run in a normal way. The attained gain ripple was 0.17 dB.

The second part of this thesis is focused on Raman amplication in metro and access

networks. The studied applications were focused on a specific network architecture,

the hybrid WDM/TDM PON SARDANA network in rural scenarios, where longer

distance links are used. Due to the network features, hybrid Raman/EDFA was

considered in the WDM ring in order to enhance the gain bandwidth enabling possible

solutions in the C+L spectral bands. Thus, several amplification strategies were tried

in order to extend the bandwidth to C+L and have an equalized gain of the dropped

channels.It was verified that the optimization of the channels dropping order and

span lengths of EDF is a good approach for the gain equalization and an optimization

strategy was proposed.

The use of amplifiers in WDM ring or remote nodes brings additional issues

decurrent from the dynamics effect on amplifiers. The latter arise from the add/drop

of channels and the use of bursty traffic in the upstream direction. These effects

can be very detrimental, especially in EDFA that have longer timescales. Thus, the

144

6.2. Directions for future work

dynamics effects of amplifiers, EDFA, Raman amplifiers and hybrid schemes of both

were studied and characterized experimentally for burst mode transmission in the

context of WDM/TDM PON. Two mitigation solutions based on optical gain clamping

were also proposed. The first one, using the saturation feedback signal reflected

on a FBG and the second one using bidirectional CW clamping as an enhanced

method. Transients in 10 Gb/s WDM ring PON using Raman/EDFA amplification

were also studied experimentally being their impact on the system performance

assessed quantitatively.

6.2 Directions for future work

The research topics approached in this thesis are important and the obtained results

relevant, but some subjects could be explored more thoroughly. Thus, from this work,

several investigations can be conducted.

Regarding the topic of the simulation of Raman propagation equations, our

attention was focused on numerical or semi-analytical methods. However, analytical

approaches could provide deeper insigths on the solutions dependency. Thus,

methods based on the linearization of the nonlinear part of Raman propagation

equations using series such as Adomian decomposition method, could be adopted.

Although the work of this thesis used an optimization strategy based the Genetic

algorithm to perform the allocation of pumps, other heuristics or combination of

heuristics could be approached, namely particle swarm optimization or ant colony

among others. The optimization of hybrid Raman/EDFA amplifiers can also be

approached using the method that was proposed in this thesis. In such a situation,

EDF lengths could be optimized and the number of Raman pumps could be

minimized. This topic is quite open to development because, since EDFA are still used

in telecommunication operation, solutions of this kind are potentially economically

valuable, especially when bandwidth extension is intended.

The work accomplished in this thesis regarding the optical transient mitigation

was thought on schemes that use clamping signal to saturate the EDFA, however,

dynamically controlable variable optical attenuators could be used to obtain the same

results. This topic was approached in this thesis through a joint interaction with

Multitel, Belgium, using a Dynamic Gain Equalizer (DGE) prototype. However,

communication issues with the device (only possible by serie port) had disabled

dynamic application in smaller timescales.

Studies on analytical description of Raman transients using a derivation analogous

to the one devoloped by Sun et al would extend significantly the results obtained

experimentally, namely for Raman amplified bursts. The latter could also be explored

more thouroughly especially under saturation.

Finally, the optimization method suggested for WDM/TDM PON for dowstream

145

6. Conclusions

traffic (the equalization of the dropped gain) could be extended for upstream. In this

situation, the optimization function could be the OSNR. Thus, a complete optimizer

for this particular metro-access network could be achieved.

146

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